It's hard to say without knowing what country you are legally a resident in and what visa type you hold for that country, but according to the European Union:
Concerning free movement, third country nationals with a long-stay visa are put on the same footing as third country nationals holding valid residence permits: They will be able to move freely for up to three months in any six-month period within the territories of the other Schengen states.
So basically: sort of, but not the way you were thinking (but if anyone has any up-to-date information from the EU regarding third-country nationals, I'm happy to be corrected!). Prior to becoming a dual citizen, I held only Canadian citizenship and had a visa for the Netherlands that allowed me to freely look for work and/or freelance. The conditions of that visa were stated clearly to me: I can find as much (or as little) work as I want to in the Netherlands, but, as I was not a European citizen at the time, I was emphatically not permitted to just pick up and go to another country. This was a bit of an issue for me at the time, as my partner had moved to Brussels and I was trying to find work there as well. Logically enough, a Dutch visa only lets you work and be a resident in Dutchland As a result, I was informed by both Dutch and Belgian authorities that I was allowed to spend up to three months out of the last six in the same member state.
I would avoid making like-for-like comparisons between the EU/Schengen/EEA states and the United States, as it really isn't the same if you are not a citizen). Being a resident in Country A doesn't mean that living in Country B would make you a resident of the latter: it means you have violated the terms of your visa and could risk deportation. Considering you aren't a European citizen, it's best not to try your luck!
It's also worth considering that things like "trying to get some kind of welfare" is actually the right of anyone who has a valid permit to actually reside in the country they're living in. If you're earning money and paying taxes for government programmes in a European country, then you should be registered legally at an address so that you can receive any benefits that you would be entitled to (like ecocheques in Belgium or vakantiegeld in the Netherlands) based the taxes taken from your income.
But, again, I think we could probably help a lot more if we knew about what country you're aiming to be a resident in. I'm no immigration lawyer, but I can at least offer my past experience as a non-European!