How do "trustless third-party watchers" work in Lightning Network?
I suppose that when you refer to "third-party watchers" you mean watchtowers. I do agree that 'trustless' is probably an ambiguous word in this context. At the time of writing, the only actual proposal with a protocol that I am aware of is PISA, by Andrew Miller et al. PISA allows users to select a custodian that will be incentivised in publishing. Users pay the custodian, and the custodian loses funds if it does not perform (i.e. if the counterparty node manages to publish an old state).

At the moment, seems like Roger Wattenhofer is working on it too. However, They have not proposed a protocol yet. Their idea is to incentivise through rewards when channels are 'saved', instead of through penalisation when a watchtower does not save on time. 

Both sides have pros and cons. 


The con of Miller's proposal is that one must choose the watchtower, having to trust in it. The counterparty might try to bribe the watchtower, so that they both earn more than the penalisation. 
Wattenhofer's proposal does not have this problem, as anyone in the network can retrieve the reward, and thus everyone is the watchtower. The problem of this proposal is that there are incentives for people to attack other people and prevent them from publishing themselves. 


This is a work in progress. As for a self-claimed trustless watchtower, there has been some discussion in the ligthing-dev mailing list here and here, but it is yet a work in progress.