The question can be reduced rather easily by stripping away the context down to the bare model. You have two groups of individuals who cannot physically affect eachother, sharing information. What happens? As it turns out, we have some pretty reasonable models of this in the world of soverign politics. We can use those to make good guesses.
In the politics which govern nations, one of the key principles is the respect of the sovereignty of the other nation. You cannot force another nation to do anything against their will. When you do, it's typically an act of war. We have a similar situation here, because there's no way to actually physically affect the other group. The only difference is that in sovereign politics, there is trade of goods, while there are only ideas transferred here.
One major limitation is that we can't assume anything the other side is true in our world. We don't know exactly how similar the worlds are, so they may give us valid truths in their world which are invalid in ours. There are also major issues here with linguistics. The famous Twin Earth thought experiment dives into the sorts of linguistic and philosophical challenges which occur.
So what can we expect, based on this. We should expect to see a sovereign nation style approach, where both sides attempt to use influence across this thin thread connecting the worlds to gain power. We can also expect a large contingent of individuals who are willing to probe the deep philosophical issues. I would expect a lot of the initial interchange of information would be done in an academic manner, freely disseminating information for both sides.
In the end, however, the regulations and transactions you ask about will be dominated by a factor which you have not given us: how the communication works. If the communication can only be done by major nation-states with massive R&D funding, you will get an extremely different result from what happens if any Joe on the street can go to the local Radio Shack and get a cross-dimensional radio to talk to people on the other side. In the end, I expect your answer is 99% based on this detail, and 1% based on human nature.