Do they pay tax in Belgium, their country of origin or a special E.U. tax?
They only pay a special E.U. tax.
As a European civil servant, your salary is not subject to national income tax. Instead, salaries paid by the Commission to its officials are directly subject to a Community tax which is paid directly back into the EU's budget. This tax is levied progressively at a rate of between 8% and 45% of the taxable portion of your salary. An additional special levy is in place until 2012 (see Article 66 of the Staff Regulations ).
No, they don't pay any local taxes on their income and there is an excellent reason for that. Such a tax would effectively be a subsidy from the EU budget and EU member states to the states that happen to host the main institutions. It could also provide those states with leverage over these institutions beyond what's provided for in the treaties.
Incidentally, this is an old and established principle that applies to all international organizations.
What I don't understand is why the EU established this “Community tax”, which seems a purely circular thing, coming from one part of the EU budget to return directly ti the same budget.