There’s probably no “electronic market” to publish offerings. I’m a GS trader with good friends in MS and UBS, I will get weekly(!) inventory feeds from them, perhaps an ftp, a spreadsheet etc. Then we agree on the terms  over phone. No automatic agreement! After agreement, borrower can request to borrow over web/ftp.. and delivery can be automated.
At its core, SL system keeps inventory, stocks lent out, stocks borrowed.
 Unlike repo, market price isn’t relevant. If I borrow 100 IBM, i agree to return it x days later. No buying! Repo involves buying ie (temporary) change of ownership.
Above is for stocks. For futures and FI instruments, there are different systems.