Can a Ph.D. student visit another lab for a week to learn a specific skill/topic? (inside EU)
Hosting another researcher is not a lot of trouble, assuming the lab doesn't require any high biosafety credentials. The host needs to provide access, consumables, and some space. The bulk of the paperwork would be done by the host, as they know what they need.

The usual procedure for PhD students is for your professor to talk with them, but you seem to be an established researcher, so if you know them you may ask them yourself. Remember that for them, their biggest commitment is the time to actually train you, so you have to somehow make it worth their while. You could offer an exchange or a collaboration, for example.

If the procedure is of sufficiently wide interest, you may convince them to organise a school and train several people at the same time, for a fee (in computational subjects, where we don't have to pay for reagents, that is usually a few hundred euros per week, depending on their independent funding).


  (I am willing to commit to travel and accommodation expenses from my own budget).


Try not to. Your institution is about to gain a very useful new knowledge, they should definitely pay for it. Also, you could probably get course credits, if you need them for your PhD.