Why would a Tx's inputs not be assumed to be owned by a single service?
Because it's possible to only partially sign a transaction (for example only one input) and then give that unfinished transaction to another party, who can then finish the transaction with their signature (for another input). Parties do not need to reveal their private keys to each other at all.

Only after all parties signed can the transaction be sent to the bitcoin network. A partially signed transaction would always be refused.