Hi Julie Born... and added a bit to your thread title to give a little context and to 'bump' it up the list!

To be honest, i think you may need to seek advice from a specialist Solicitor for this one as there are so many considerations. As a general rule, it is the law that applies in the country the work is performed that applies. So you could have Hungarian immigration, tax and employment laws to consider. But if they are paid from the UK and classed as an employee in this country too, then our immigration /right to work and tax legislation would apply.

I think i would want to get some really clear legal advice on this if i wasn't sure.

Employment law is territorial in most countries. Thereforefor work work carried out in Hungary will fall unders Hungarian statutory provisions. If you also have a UK contract claiming UK law, then this will also apply. when the two contradict each other, the more favourable of the two will apply
Taxation in most countries is if someone is resident and working in that country, then that country's tax must be paid. You must also comply with any Hungarian PAYE and reporting obligations.
Social Security in Europe nomally requires employer and employee to pay SS in the country where they are living and working. An employer of a person who is temporarily posted to another country can obtain an A1 certificate to kep them in the social security system of the employer, provided they were already in this system and provided it is done before the temporary assignment starts. If you want to put him in UK SS, you'll need him to work in the UK for a few months before you are able to obtain an A1 certificate of temporary secondment.
Immigration - depends tremendously on the country of origin, nationality of origin, and country of destination. Can't help you with this on and you'll need specialist advice on this
If you have a local company in Hungary, the simplest route for all concerned is to employ them locally and put them through the payroll of that company. If you don't, then your options are either a local body shop, or using a good legal/accounting firm to handle contracts, labour law, payroll, employer reporting obligations for you (likely to be expensive)

In reply to Steve Bridger:

Thank you Steve.

In reply to Teresa:

Thank you Teresa.

In reply to Ray:

Thank you Ray, that is really, really helpful.