It generally doesn't require input from the target network being bridged, unless you're running a large public bridge like the ones we host on matrix.org. You do however currently have to run your own homeserver in order to run your own bridge (although in future we're looking at ways of getting around this for 'personal' bridging purposes).

If you run your own homeserver, and your own bridge to Rhizome/wherever, you can think of the end result being a bit like bitlbee or a pidgin-in-the-cloud... but decentralised, and with the richness of Matrix's conversation semantics (e.g. arbitrary data types) rather than being limited to IRC.