Can a researcher get his full salary from a European Project?
So, after quite some time spent on reading the paperwork and talking with helpful staff, I finally understood how this system works. There are different amounts involved: 


S: the salary of the researcher, corresponding to the gross amount, i.e. before tax. For instance, 50.000 euros per year. 
r: the overhead asked by the host institution, which basically corresponds to the indirect costs of hosting the researcher (renting the office, paying the admin staff, etc). This is an agreement between the host institution and the european commission, usually calculated as a percentage of the salary. For instance, where I'm working now, it's 67%. In some places, it can go up to 100%. 
T: the travel and equipments expenses. It basically depends on the expected number of conferences to attend per year, the meetings organised for the project, etc. For instance, based on 4 conferences per year, 2500 euros per conference, that makes 10.000 euros. 
C = S + rS + T: the total cost of the researcher per year. For instance, that would be 50000 + 33500 + 10000 = 93500. 
Req = 0.75 * C: is the requested amount to the commission (assuming the host institution belongs to: profit public bodies, secondary and higher education establishments, research organisations and SMEs, 0.5 otherwise). So, for the example, we would request 70.125 euros per year. 


Note that the request amount is 10.000 euros above the gross salary of the researcher (deducting the travel costs). In other words, one could consider that the 25% the host institution has to pay are indirectly paid by giving an office and proving staff support (who, in general, would be paid anyway, EU project or not), and therefore it does not necessarily mean that the researcher has to find another funding source for the missing 25%.