Most countries won't issue you a work permit without a lot of paperwork, cost, and time. Think 'send your passport off to the embassy for 3 months' type hassle...
And most employers don't want to employ someone abroad. Employment law varies widely by country, and unless they already have an office in that country they don't want to do all the tax and reporting stuff for only you.
Long term abroad in one country might be doable, but 30 days in each country is pretty much impossible.
Of course, if you were a rock star coming in to perform, they'd definitely want their cut even if you were only performing for a day. The laws haven't advanced to cover this situation entirely well, so there's a lot of leeway on what you can get away with.
If what you really want to do is live and work in one country while pretending that you live and work in another one, I don't see why either country should facilitate that.
It's easy enough to do that "properly" too, form a company or self-employed tax status in the country you are living in, and invoice the company in the country you work for. Then they aren't in a compliance issue and you are paying taxes where you work and live.
What's the problem?
only if they offer something like a digital nomad visa. otherwise, usually you are not allowed to do any work at all on a tourist visa, even if it's remote work to your home country. nobody cares if you check your work email or do some small stuff, but technically no work is allowed.
form a company or self-employed tax status in the country you are living in
in most countries you can't do that as a foreign citizen unless you have a proper visa for that. you can probably open a company in a foreign country, but you likely still need a work visa to actually work for your own company. otherwise anyone would just go to open a company just so they can live and work in that country.
It's true you can't necessarily do what you want, where you want. That's fine though, no?
> unless you have a proper visa for that
Yes, that is part of doing it "properly", I should have been clearer.
My point was that there are mechanisms for doing all of this stuff; someone may not like the mechanisms but that's not the same as them not existing.