Working for one EU country and living in the other [closed]

From an immigration perspective, you could simply reside as an economically inactive person. Unless your salary is very low indeed, it should satisfy any income/ressource requirement, if you even need to register officially anywhere. The only thing that could be more complicated is health insurance but that too ought to be solvable with some money. Your presence is in any case fully legal and you do not need a permission to work.

Being considered a “worker“ is attractive because it exempts you from the income/ressource and health insurance requirements and protection from expulsion is very strong indeed. If your work is “genuine and effective” (even part-time), you qualify. If you have to register somewhere, the most you will ever need is a passport/ID, work contract and perhaps a proof of address (as required locally for citizens). You do not need to document savings, income or health insurance. Historically, this was the original scope of the EU (then EC) freedom of movement.

The “economically inactive“ category was added later to extend freedom of movement rights. However, you don't need any permission to work or engage in other economic activity. Economically inactive people are definitely allowed to do that. It's just a less permissive status that comes with additional requirements (namely having — minimal — ressources and health insurance), if you are not able to provide a work contract or anything else to prove you qualify as a worker.

None of this covers tax law or mandatory contributions to social security systems, which can get very complicated.

As several others have pointed out, there are almost no limitations on registering your residence in any EU country. As long as you can show you physically have a place to stay and means to support yourself, your registration can't be denied.

Either way, the key here is to get your employer on board. If your employer is not going to cooperate, there's nothing you can (legally) do. But in that case, I suspect that your employer would also take a dim view of you not showing up at the usual workplace for months or years.

There are several avenues that you and your employer can take to get the paperwork sorted out:

• Get posted to wherever you wish. This only works if there's a plausible explanation of that (e.g. a customer or partner in the other country that you need to work with). Expect the relevant social insurance authorities processing your A1 to thoroughly scrutinise whether your posting isn't a fake one (trying to circumvent the labour law in the host country).
• Amend your work agreement to say that the place of work is going to be in country X (Y, Z…). Your employer will have to register with the authorities in these countries and start withholding income tax and relevant national insurance contributions. There are many agencies that will sort all of that out for you. Just ask any major international accounting/tax network and they'll send you a quote in no time.
• Go self-employed and replace your employment agreement with a sub-contracting one.

In all three cases, you'll need to apply for an A1, but that's a straightforward process that won't take you more than an afternoon. The authorities will then sort it out between themselves and tell you which country you belong to for the purposes of health and social insurance.