I am wondering if (and how) a Ph.D. student could learn a skill that is the area of expertise of another lab inside the European Union, but in a different country than the university on which the Ph.D programme is currently done.
So lets say my lab does not have any experience in X procedure. I know that university in another country inside the EU has expertise in X.
Learning the basics in X takes about a week of hands-on lab experience.
How could I initiate the conversation between universities to get me to the other lab? is this even necessary as I would not be paid for that week while visiting the other lab?
Or do I only need to get the permission of the lab supervisors and do not bother about the paperwork.
I am willing to commit to travel and accommodation expenses from my own budget.
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.