Let's talk about some of the common chemical types used in life:
- Amino acids
- Other factors
- Putting it together
As mentioned above, both Sugars (the chemicals used for energy transport and storage) and amino acids (the chemicals used for cellular structures) are chiral (meaning they have left & right handed types). Use the above answers to see what handedness matters.
At the worst, the odds of using the same sugar chirality is 50% (some experiments have shown a preference for one handedness over the other in some reactions).
If non-terrestrial biology uses sugar in its energy storage and transport mechanisms and we share chirality, then we should be able to absorb and utilize the sugar from non-terrestrial organisms.
Another aspect of amino acids is that I read somewhere (but can't recall where - so no reference) that terrestrial biology uses 47 of the 50 amino acids most commonly found in nature. Meaning it isn't a stretch to suppose that other biospheres using amino acids in its biology will likely use most of the same amino acids.
As above at the worst, the odds of using the same sugar chirality is 50%.
If they also share chirality, then we should be able to break down and absorb at least some of those amino acids.
Terrestrial biology uses DNA and RNA to store and transmit genetic information. In terrestrial life, DNA uses 4 nucleotides and RNA also uses 4 nucleotides. However, RNA uses Uracil in place of Thymine in DNA. That means terrestrial biology uses a total of 5 different nucleotides.
I'm not a biologist but I assume that other nucleic acids would serve the same purpose, however, those used in terrestrial biology out-competed all other methods of storing and transmitting information over the last 4.3 billion years.
I assume this is both due to it's ability to preserve that information as wells as providing an extremely small probability of mutation (which gives the species a chance to adapt to other environments).
Let's assume chirality isn't a factor and the non-terrestrial biologic uses at least some of the same sugars and amino acids. There are some other interesting things to explore.
One that I found fascinated was the fact that much terrestrial animal life uses a protein analogous to hemoglobin to transport oxygen.
In octopi and horseshoe crabs, this protein is called hemocyanin and uses copper instead of iron as the critical metal ion.
Other metals used in hemoglobin analogs are vanadium and manganese. However, all three of these analogs are much less efficient than hemoglobin (<1/4 for the best).
Another thing to consider is that terrestrial life develops to survive in a specific environment. Freshwater fish can't live in sea water (it's too salty). Salt water fish can't live in fresh water (not enough salt).
Organisms which consume plenty of a vitamin through its diet, lose the ability to produce that vitamin (e.g. human bodies can't make vitamin C).
Elements become toxic to organisms which are unused to exposure to that element (e.g. the arsenic comment above).
Putting it together
No one knows what alien life will be, how it will function, what chemical processes it will use, and whether that chemistry could digest ours. However, if the non-terrestrial life uses process similar to that of terrestrial life (which isn't entirely improbable) then we can make some guesses.
If we both use sugars and share chirality, then terrestrial and non-terrestrial organisms can digest each other's sugars. Otherwise we can't.
If we both use amino acids and share chirality, then terrestrial and non-terrestrial organism can digest each other's amino acids. Otherwise we can't.
Organisms from one biology are unlikely to be able to co-opt, infect, or otherwise bother the cellular operation of each other directly (no viral cross infections).
However, organisms from one biology may happily set up shop in the body of organisms of the other biology (e.g. alien bacteria living in our mouths). It may take a long time for our own immune system to adapt a response to shutdown those opportunistic colonies.
For planets with different elemental abundances, we may find that common composition of the organisms contains enough of certain elements to be toxic to the other. I would expect that in most but not all cases, this would mean the toxicity goes both ways. However, in certain special cases this might not be true.
So if you're writing a story of the type you outlined, then you could sprinkle in some of these factoids and then make comments that it is highly unusual for two biologies to have an asymmetry in toxicity like the one you describe.
Even if it turns out to be unlikely it's certainly possible and know one knows whether it is truly plausible or farfetched.