(I request that this answer may be viewed as an addendum to Pete's nice answer here, in light of Nate's comment above.)
As Nate pointed out, most of the typical referee responses would be concerned with stuff that should enter the manuscript. So, assuming that all his useful suggestions were incorporated in the text itself, there isn't generally more meat from that conversation that could warrant a separate 'response log' to be uploaded anywhere (I'm assuming on arXiv, for example).
But the impression that I get from the question is that, OP is inquiring about suggestions that go deeper than the above paragraph. In some cases, it is possible that referee queries stuff on the lines of
How is [a fact that you established on the basis of your calculation in the manuscript] consistent with [a sacred tenet, or a well-established or experimentally verified result] ? Aren't the two incompatible because of [some qualitative reasoning, devised by the referee]?
The reasoning looks valid to you, so you sit down and calculate the implications of your calculation on the established fact, and find that the two are indeed compatible. Then, you identify a weakness in the qualitative reasoning, and let the referee know about this. Now, all this isn't worthy of being included in the text of your manuscript, since it is off-track from the overall theme of the work. Yet, this is a valuable piece of information, and is likely to help future readers because they may also reason this apparent contradiction. Responses of this kind are worthy of being put up. Occasionally, one encounters those one-page or two-page ''Comment on [a paper]'' sort of things on arXiv, so these can definitely be put up too. It doesn't necessarily have to be journal article manuscript always.
Lastly, regarding acknowledging the referee, there are two options - either take their permission (ref - Pete's answer), or you simply acknowledge ''the anonymous referee'' for pointing it out, in case option 1 doesn't work out. I know some instances where this has been done in my field, but one example that I can find is over here. Sorry, there isn't any corresponding arXiv version for this, so if you can't access it directly, here's the relevant excerpt:
The author would like to thank the anonymous referee for making insightful comments which have been helpful in improving and updating the manuscript.
But seriously, option 1 is the better option (why strip the poor guy of his due credit!).
Hope that helps.