Unfortunately, while it does keep your history, the search is pretty lousy (there is a better search available for paying customers) and you can't export private conversations (unless you pay them).
They gotta make money, and I'm not paying them, so I understand withholding features. That said, I keep looking at the Matrix ecosystem, and I think the only thing keeping me at MM is the switching cost at this point (not being able to export messages is an effective lock-in feature, though rather user-unfriendly if you ask me).