Besides Beeper, the only consumer bridge hosting service I'm aware of is Element One, but it's only barely maintained, so I'd definitely recommend self-hosting over that.
Messages are stored in encrypted form on the Beeper server and the Beeper client has a local search index (the same one used by Element desktop: https://github.com/matrix-org/seshat)
Besides Beeper, the only consumer bridge hosting service I'm aware of is Element One, but it's only barely maintained, so I'd definitely recommend self-hosting over that.