I think it’d be pretty fast, but maybe not imperceptibly.
Take a look at the first gif in my update here showing the bug flying between tiles.
That is done with a similar handshake to what you describe. The original tile has a bug that reached the “edge” of the tile. The tile sends a request to transfer the bug to its neighbor. The neighbor receives this, decides it has room to accommodate the bug, and turns on its bug code. It also transmits a “I took the bug” message back to the originator. That causes the bug in the original tile to turn off.
The gif isn’t the best quality, but you can briefly see frames where the bug appears in both tiles while the transfer is in progress.