I can only speak about how my company uses Skype (which may differ since I don't know the ins-and-outs) but my conversations end up in my exchange inbox in a folder titled "Conversation History".

So at least, for me, Skype for business has this functionality.

Personal skype stores your chat history local to the device- I found that out recently when I was looking at my non-existent conversation history on my new phone.