Can a non-miner third party help other people's transactions to confirm faster? If I recall correctly, in 2017, when the mempool was full, there were some guys bumping other people sent transactions that were stuck in the mempool.

Is that possible? How does that work?
No, in general non-miner third parties cannot influence the priority of transactions except by outbidding them.

Unconfirmed transactions can only be reprioritized by either the sender or one of the recipients:


Any recipient (including the sender if there is a change output) can craft a child-pays-for-parent transaction that will incentivize the inclusion of the stuck parent.
The sender could publish a new version of the transaction, if it was flagged as replaceable (via BIP125) upon creation, or if they attempt an outright doublespend.


The service you describe was offered by mining pools. These mining pools would promise to include the transaction in the next block they'd mine in exchange for an out-of-band payment.