This issue replaces paritytech/grandpa-bridge-gadget#20 and tracks implementing #3 below:

A node running BEEFY worker/gadget, generates/imports and stores BEEFY justifications as follows:

1. when synced, it participates in voting gossip and builds BEEFY justifications itself from gossiped votes,
2. when behind, it imports BEEFY justifications alongside imported blocks (as part of block import - added in Lean BEEFY to Full BEEFY - don't skip (older) mandatory blocks and import justifications #11821 and beefy: initialize voter from genesis and fix initial sync #11959),
3. runs dedicated BEEFY justifications sync protocol as a libp2p RequestResponse protocol - where node can request BEEFY justifications for certain blocks from connected peers, and also answer any requests it gets.

#3 above is required when syncing from a peer that might be missing some/all BEEFY justifications.