The answer is no, its not up to you. Its their income, so they pay their own taxes. They are not your employees, they are contractors. As contractors you dont have to pay any SS or Medicare taxes. Your tax burden is ZERO dollars.
The IRS in the states has strict rules about who is a contractor. Make sure you review those before hiring these freelancers.
Last, unless you are hiring Companies to do the work, you are required to have them fill out paperwork so that you can file your 1099 at the end of the year. Otherwise you are not able to write off the expenses of hiring these folks or will face penalties.
Generally, you leave it to them. In the US - you have to file 1099's on behalf of any contractor you pay more than 600 to. You'll have to see what the local laws are for the UK and EU - I had freelancers in scandinavia and had to pay an equivalent 'payroll tax' on top of their wages.