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This has been discussed many times here. You can search the forums for details.

In short:

You need to start a business or declare a tax basis such as KATA. For some business types, you may need to show an accredited diploma for the work you will be doing. Also, if you work for clients in the EU you have to collect VAT (or at least pay the HU tax office the VAT amount from your income if you do not have a VAT number) on services such as web design To make it more complicated, you can declare to be VAT except in Hungary, but that really only applies to HU clients. Finally, even if you do not work, you may still be liable for tax in Hungary on other non work income (royalties, dividends, etc.) -- or not if the income country source has a Tax Treaty with Hungary.

I am assuming your tax number should be still valid despite your  "no work" residence status (but you need to ask at the tax office to make sure).

Allow me to point out the difference between "tax id number" and "tax number". I'm not sure that these are the appropriate terms in english, but hopefully you'll get the idea. Everybody who registers as a resident will get a "tax id number". With this you cannot issue invoices. You can do some occasional work, though, if you have the right visa, but in your case this is not an option. To get a "tax number" that also allows you to issue invoices that many customers would expect, you have to either request it as a private person, this would suffice for some occasional work. The next step up in complexity and possibilities is registering as a self-employed person as KATA or EVA. And the final, most complex option is registering a company, e.g. an LTD.

Edit: the original, hungarian terms that I tried to translate are adóazonosító jel = "tax id number" and adószám = "tax number".

klsallee, atomheart, thanks for your detailed responses!
I don't think I am ready to go full self-employed at this point, so I was thinking more in lines of getting a full-time job while sometimes receiving gigs from that agency. Of course while minding all the signed NDAs.
So yeah, thanks for the details!
Marry Christmas and happy holidays!