How do "trustless third-party watchers" work in Lightning Network? The main point with LN is the need to monitor the blockchain. The reason for this is to detect if the counterparty misbehaves by publishing an old channel state to the blockchain. For a node running on a server, that is not a big problem since it is always online, but mobile nodes might go offline for days, so how do we deal with this issue?

The need to monitor the blockchain actually only exists if the channel is used in both directions: if you are only paying, any outdated commitment transaction will be more in your favor than the current one, so there is really nothing to do!

But in case of two-way payment channels, this seems to be resolved by "trustless third-party watchers". Since I can't find much literature on the internet about them, I was wondering, how do they work and how exactly are they trustless? For that matter, how do you actually trust that anybody is going to publish on your behalf, and why? Are there incentive mechanisms in place for this to be completely trustless?
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.