There aren't serious technical obstacles but it may be a poor idea if it turns into a hive of abuse. I got into a discussion about implementing forum software a while back and I said I wanted it to have encrypted DM's, and several operators jumped in and said it would probably attract more trouble than it was worth. People wanting direct private comms can always use encrypted email. On Reddit whenever I wanted to discuss anything the least bit private, I generally asked to exchange email addresses with the person or similar, and went from there in some cases to a self hosted encrypted chat.
Anyway this feature should be weighed somewhat carefully. Are you going to do the thing with warrant canaries? Any attempt at all to conceal metadata? Etc.