Hiring people in multiple European countries and payroll taxes

Suppose I have a company that's incorporated in France and that I want to hire a few sales people in several European countries to help sell my Saas subscription. Do I need to incorporate a subsidiary company in every country in which I want to recruit people, and then file for payroll taxes in all sorts of countries, or can someone in Germany or in the UK be an employee of the French company and thus save me the burden of incorporating and reporting taxes in several countries at a time?

Thanks.

I've been working on the basis of a consultancy contract recently, and as it happens also on selling a SAAS based project. In my experience you don't quite need to set up legal establishments in each country, as Dmitri mentioned. You do need to watch out for local finesses, e.g. the maximum duration of a consultancy contract. At start you probably are OK but watch out if you're using someone for longer than 2 years in a row; in some countries a 2+year contractual relationship may be considered as a half full time employment.
Side remark: if you're from North America and if this is your first effort into Europe, please make sure that your financial processes can cope with VAT.

No, of course not, you hire people under a consulting services agreement. You wire them the money and paying taxes is entirely their responsibility. You don't need to set up legal entities there, nor do you need to pay their taxes, medical insurance, or anything else.