It's based on Signal protocol and it's also open source. However, unlike Signal it doesn't harvest phone numbers.
https://github.com/wireapp
It has been a while since I looked at Wire, and I didn't look very deep, but here's what I noted:
Self-hosting was unavailable at the time.
I believe they violated their privacy policy a while back, by accepting new owners/investors without notifying their users. That kind of behavior is telling of what to expect from an organization, and potentially dangerous (depending on your threat model) if you're trusting them with anything, such as...
I have read complaints that they stored cleartext contact lists on the server, but I haven't verified this myself. (The first two points were already deal-breakers for me, so I didn't bother.)
Looks like both the client and server are open source now, so it's definitely possible to self host. Unlike Signal, Wire also supports federation nowadays https://docs.wire.com/understand/federation/index.html
Huh, never really looked into Wire but it seems like its a great option and now im wondering why people bother with signal or matrix if this can do both encryption and federation
Federation sounded interesting so I looked at the website and it sounds like on prem can't yet federate with people using "cloud" which I guess is the hosted version - they can only federate with other on-prem instances.
It looks promising though and would be cool to host my own instance and still chat with friends.