Considering that interplanetary trade will likely be limited because of cost of shipping vs product value, how would interplanetary money be valued against each other? I'm thinking more along the lines of tourists or even business travelers. How would you know if your 80 green Martians should buy you a burger and fries or night at the Hilton in London? How would 2 almost separate economies decide on a fair exchange rate? We used to have actual goods or physical stuff like gold or silk. Will we be back to bringing beads for the natives when we want to travel?
If there is no exchange of goods or services then there is no value for each currency in the other worlds. Currency is purely defined by what it can be used to purchase.
There for the question is a matter of what is imported or transfer between the two worlds. It's likely that some sort of exchange does regularly occur. If people are traveling between these worlds regularly then services are being exchange even if items and trade goods aren't. These services can be the basis of setting economic value. You have a someone playing taxi between the two worlds. How much would he ask the Martians to pay him for his time and buying gas on Mars to be able to taxi them to Alpha Centauri. How much would he charge the Firaxians from Alpha Centauri for a trip to Mars (+5 Internets to anyone that gets the allusion). That cost for services rendered at both worlds a good starting basis for an exchange rate between the two worlds. Judge what services can be purchased on each planet.
In addition there will be financial exchanges of items. Large expensive things like raw materials would be horrible wasteful to ship. However, a cargo hold of tiny expensive micro-processors can still be cost efficient to ship potentially. If you assume that its cost efficient enough for people to travel between planets at all regularly then the cost of travel is low enough that small refined goods can still be cost efficient. Arts and trade goods may also have resell value.
Your tourist will bring his Martian money and use it to buy things. There will be some interplanetary exchange which decides his wealth based off of goods/services exchanged between the two planets (or exchanged between planets they both have commerce with) and will regulate the prices. Honestly it's no different then today's economic system in most ways. Only three factors are different.
As you said the harder it is to ship people and resources to a planet the lower the money exchange rate will be. As a pretty accurate rule of thumb the less likely it is for someone to travel from planet A to B the worse the exchange rate will be for them. If you're assuming at all regular trips to the planet then this won't be an issue, because if it's cheap to send people to the planet it's cheap to send refined goods as well. If someone is visiting a planet that is far away, expensive to reach, and rarely visited by people from his planet he is going to suffer significantly in the money exchange, and may be better bringing 'beads' to exchange, though his beads will be art, micro processors, and similar light weight fungible items.
Unless you have some form of faster-then-light communication system in place there is a lag in communications which causes uncertainty about the financial state of the planet you're traveling from. This means planets with more stable and less dynamic economies and governments will be favored better in currency exchange, to adjust for the uncertainty of having to decide a currency exchange without knowing rather some disaster has befallen the other planet and tanked its economy. The further the planet is the more stale news will be and the worse the exchange rate. How significant lag is between the worlds, i.e. how fast does your FTL system allow travel and communications between systems, will strongly effect how significant this factor is. A lag of a few days doesn't matter much, a lag of a few years matters quite a bit!
If bullet point 1 or 2 leads to any noticeable loss in exchange rate then you may be better for someone to buy up all of planet B's currency from locals of planet A before he leaves for planet B. Since planet A will have an exchange rate that favors A's currency over B he will get a very good deal. If he is going to later return home to A from B he would also want to buy as much of his home currency up with the remains of his planet B currency before leaving to take advantage of the favorable exchange rates. This is called arbitrage, and it can be a viable way to make money (or at least lower loss of money for expensive travel). Arbitrage is also the reason that any economies with regular exchange of people will tend to equalize to a steady exchange rate, if there is a significant difference someone will take advantage of arbitrage to make a quick buck (or paso, or space-looney).
an update on a very old answer of mine. However, there is one other exchange option to consider information.
Lets look at a world that is so far away away/expensive that travel rarely happens and even refined technological goods are not cost efficient to send to the world, in which case it's likely going to be absurdly expensive to travel to the world. For now lets also imagine a scenario where no FTL comms or other means of long distance communication without travel to the world exists. What can you take as 'beads' to exchange when there is no currency exchange?
Since it's so hard for anyone to travel to the world the world your traveling to likely has not heard anything about what your world has been up to for awhile, and not gained any information from your world. Things like books, movies, video games, software, etc, all can be purchased and stored as ones and zeros on your future-thumb drive to take with you when you travel. Your arrive on the new world with Hundreds of millions of dollars worth of sellable 'products' in your pocket, with almost no expense beyond what you payed to travel to the planet yourself. These will be the best 'beads' for you to use, no one on the new world has seen "Star Wars Episode XXVII" or "Finding Marlin" movies yet, Look at the box office grosses for movies in foreign countries, and imagine the kind of money you could make if you brought an entire years worth of movies to release on an entire different planet, along with books, games, etc as well...
If your world has FTL communication something similar, meaning the other world likely isn't waiting for a traveler to bring media to them, an information based economic exchange can still be used as the basis of transferring value between two worlds; the process is simply more complicated, but also more robust.
Lets look at an overly simplified example first. Imagine your Home Planet A finally finishes Half Life 3 game, to much celebration and sales on World A. Once their done making a profit on their own planet whoever owns the rights to Half Life series in the future (lets say Google, their own everything electronic by then I'm sure) decides to look at other markets to sell their game in. Google calls up affiliates on world B and sell them the rights to publish "Half Life" on World B, in exchange for Royalties worth millions of space-bucks off the sales. Now these millions of dollars on World B are of limited use to Google at first glance, if it costs too much to ship from B to A then Google can't buy anything from B with their space-loonies; it's hardly worth the crypto-card the money is published on right? However, Google already produced Half Life 3, it costs them nothing to sell publishing rights to World B, so a few million world B space-loonies in royalties that they can earn with a single expansible call still sounds potentially useful, their figure out something to do with it later.
Now, imagine you are traveling to World B, something prohibitively rare. You call up Google because you hear their so rich they haven't settled for monopolizing our world and started in on owning part of World B's economy s well. You offer Google five hundred dollars in exchange for a few hundred thousand space-loonies worth of money on World B, five hundred dollars of money on their home world, that they can actually spend to buy things with, is worth allot more then a hundred thousand loonies trapped on a world where they can't access it, so Google happily takes your 10 thousand dollars and calls up World B and tells them that they should give you a hundred thousand dollars worth of royalties when you arrive in World B. Without carrying a single physical item you've manage to buy up enough money to be rich as soon as you land on World B.
Of course in reality this is overly simplified, because Google would have more to do with their space-loonies then wait for a traveler to World B. Maybe when Google sold the rights to release "Half Life 3" to world B they agreed to accept half their usual royalties in exchange for publishing rights to the "Duke Nukem Infinity" game that world B just finished producing. Now both worlds have gained something sellable on their own world, and thus of real value, via an economic exchange with the other world. Constant buying & selling of media, and other forms of information. from the two worlds like this can create a thriving information economy without any travel between the worlds. It could be you already have exact exchange rates, markets for buying and selling currencies, even integrated stock markets with people buying futures on other worlds currencies etc long before you decide to travel to world B, you need only buy exchange currency on the market based off of the exchange rate decided/controlled by the information economy. In a way this would be bad news for you though, you won't get as good a value buying the other planets currency if there is already a strong economy, and if travel really is that expensive then the lack of some form of arbitrage to help recoup travel expenses could make the trip bankrupt you.
As with other answers, I point out that you need to separate buying power from numeric value. As for working this balance, I follow a school of thought my father invented, which is named, tongue-in-cheek, "Beeronomics."
In all cultures that allow the imbibing of alcohol, beer is present. Workers generally need the same degree of non-sobriety the world around. The exact degree is unimportant, other than that it makes up a large enough portion of a worker's disposable income and can be treated as a necessity good for purposes of production and pricing.
To determine the exchange rate (for tipping purposes), simply look at the price of the local domestic swill, and measure your tip in beers. So if you would have tipped \$10 at home, and a beer is \$3 at home, you should tip \$5 in a foreign location where a beer is only \$1.50
It is, of course, a very rough concept, and should never be thought of as anything more than that. However, it is a powerful tool for making exchange rates make sense. It can be applied quickly when you do not have time to do the multiplications to get the real exchange rates.
If you identify any common needs between Martian and Earth economies, those common goods can be the metric you use from which to build your exchange rates.
If there are no common goods, that may suggest that Martian Wahoos and American Greenbacks might not actually be exchangeable, because they actually measure two unrelated markets with no opportunity for trade between them. This would trivialize your question.
To trade within a single interstellar empire in a single economy, you can use fiat money (not based not precious metals) - Chinese started using them in 11th century.
Between empires, you would rely on clearing houses and/or banks who do trade with both sides. So you deposit your valuables in a branch in one empire, and collect equivalent in another empire. Banking started around 14th century to solve exactly these problems. First foreign exchange contract (we know of) was in 1156.
Obviously, any such direct arrangement assumes frequent travel between participating planets. If travel is slow or complicated, there might emerge arbitrage making profit from inefficient markets (so bigger part of price difference will go to the arbitrager).
With even slower travel and higher cost, only the most expensive/profitable material would be transported. Or, Transport would be so slow, that generation ships will be hopping to next system and no commerce would be possible. Commerce depends on cost per pound per year in transport (and risk of loss of cargo) so depending on your parameters, different settings are possible.
If transport between the two planets is extremely expensive, this would force each planet to be self-sufficient as far as basic life requirements are concerned. You would have a hard time signing up colonist for a one way journey to a planet that required monthly shipments of water and oxygen.
I would therefore conclude that currency from a different planet would only have value as a novelty item, to be collected by affluent residents of other planets specifically because it is rare. The multi-millionaire that steps off of the space-liner might be in for a big surprise if his pockets and credit accounts are full of foreign currency.
As for official transactions between the governments of both planets, specific negotiations would have to occur for every trade at first. Eventually, each government would possess enough of every other government's native currency that commerce could be practiced in the currency of the product's origin, allowing items to be purchased at prices established by their local economies. With shipping costs making up the vast majority of any item's cost, minor fluctuations in the purchasing currency could be ignored. Ultimately, the laws of supply and demand would rule all.
One currency would span the void without excessive shipping costs and that is knowledge. Discoveries on either world could be sold to the other again under the laws of supply and demand. Those transactions could either be performed in the economy of the knowledge owner or in that of the purchaser, depending on what associated purchases of physical products might also be occurring at that time.
Unless we found another intelligent species on another planet to trade with I assume that we would be trading with humans, that originated from Earth.
In that case we would probably begin by using Earth currencies until infrastructure is set up on another planet. Eventually us people on Mars will come up with our own currency, as we get paid more on Mars, the price of living and food is much more expensive, travel is more expensive, and eventually the exchange rates should line up, with the average price of living matching up with earths.
If we are trading with another intelligent race, I'm afraid we won't use currency at all, at least to begin with. If this other race has a currency already, it would be worthless to us at first, but the more our races mix, people traveling to and from this other planet, the more the currency will be used by humans and the more accurate exchange rate will be possible. We need to build up an economy based around the currency before it can be used to trade with. But I believe trading with aliens will just be an exchange of goods, and the exchange rate of that will just be based of
You might want to read Neptune's Brood by Charles Stross. It features details of interstellar banking, wealth generation, and currency. The main characters are accountants, and the main plot concerns a financial vehicle.
In our own realistic future, transport on and off gravity wells like planets will overwhelm the price of goods. Basic elements might be cheap from asteroid mining. So what would be valuable to trade?
There was a scifi, whose name escapes me right now, that basically said that you can't afford to ship water to Mercury, even though Mercury has none (or next to none) and the Oort cloud (or Europa, etc) has plenty... because of the time it takes to ship it. The cost of your money sitting around for 5 years (or whatever) makes it unprofitable to do anything that long-term - even though it needs to get done.
There will be great lag in communication due to the distances between the purchasing value of two currencies distanced by light years. (by the way great arbitrage oportunities for space craft staying in the midle and trading the goods). Unless you use alcubierre hyperdrive spaceships at many times the speed of light, then prices can be have less differences
Take advantage of Quantum Entanglement. Instantly transfer and track all transaction anywhere in the universe.
"The use of quantum entanglement will make it possible for all participants in a network to simultaneously agree on a measurement result without a proof of work system."
Interesting article: https://arxiv.org/ftp/arxiv/papers/1711/1711.04235.pdf