I'll say from the jump that I am getting most of this from Koenraad Verboven's The Economy of Friends: Economic Aspects of Amicitia and Patronage in the Late Republic. However, when I was searching for a term I stumbled on this article [pdf] which I think does a very good job of summarizing the issue and the difficulties of interpretation focused on tax farming organizations.

This is a difficult question to which the best answer is going to be something like "kind of". I think maybe the easiest way to describe it is that Roman law recognized complex partnerships that took on aspects of corporate personhood but did not quite rise to the level of being a separate entity from the people actually involved in the partnership or existing outside of the venture. As an illustration I will draw from chapter 14 of Justinian's Digest, which is primarily concerned with maritime law and I find useful because they are very complex enterprises that require a degree of shared risk and action but also asymmetry of investment (eg, one person has the ship, one person has the capital). The "Rodian Law" concerning the jettison of cargo is a good example: what happens if, during a storm, one person's cargo has to be jettisoned in order to save the ship? In that case, the person whose cargo was jettisoned is entitled to sue the captain (shipowner) for damages, and the shipowner in turn is entitled to collect from the people whose property was saved so that the loss is evenly distributed. This is regardless of whether the goods were light weight and thus could not have been jettisoned (like jewels), meaning the venture as a whole was considered, in a sense, a whole rather than just a set of individual parts. However, the actual mechanism through which this was achieved was personal relationships--the owner of lost cargo to the ship captain, the ship captain to the other cargo owners. If you can, I would recommend glancing through that chapter just to get a sense of how deep in minutia these questions could get.

This said, there were institutions that were much closer to modern corporate entities, most notably collegia ("trade association" rather than "guild") and temples. However, these were quasi-"public" (or "political" rather) in that they were formed through government action even if they were not public institutions. There are examples of mercantile organizations associated with particular cities or municipalities (for example, the merchants of Tyre in Puteoli) but I have not seen evidence that these were actual corporate bodies rather than social clubs.

I hope this answer makes sense, because there is not a straightforward "yes or no".