This is really /u/missingpuzzle's jam, but it is a topic I bump into. There was a substantial degree of contact, but the specific forms of contact you are curious about are significantly more difficult to discern. For example, while we can imagine that large emporia like Muziris in India and Khor Rori in Oman would have this sort of multicultural nature, we can't really prove that, and even in the best of circumstances excavating evidence of trade languages is extremely rare. We often implicitly assume that such multilingualism was at least present, because how else can trade function?, but saying more than that is impossible.

That said, there is a growing understanding of the multivariant nature of the western Indian Ocean commercial sphere at this time, which more or less consists of six different nodes: the East African coat (which can be further subdivided into Axum/Ethiopia/Somalia and "Azania", likely Tanzania), Rome, the Arab peninsula, Persia, and India (which can be subdivided into whichever groupings of the coast you like, usually northern, the Western Deccan, southern peninsular/Tamil, and Sri Lanka). Of these, we have a reasonable understanding of Roman merchant activity and rather less of Persian activity, but the others are comparatively obscure. It was not too long ago, in fact, that it was assumed that only Roman vessels were capable of making the transoceanic crossing using the monsoon winds, partially because maritime ethnography really only began after the advent of the steamship ended all "indigenous" upper level cargo ship construction. We now know, however, that the trade routes of the Indian Ocean have enormous time depth, so often time we know that there was exchange but it is difficult to characterize it. Even in the case of the exchange between Rome and southern India, which is certainly the best understood, questions of cultural exchange are highly contested.

My perspective on this is very much from Rome, but Himanshu Ray's article in Ships and the Development of Maritime Technology in the Indian Ocean should be enlightening. If there is a specific aspects of the commerce you are curious about I may be able to help you, particularly with more economic points.