I don't know the rules in Europe, but where I am, a company or an individual can hire someone (a pilot) to work for them much in the same way someone would hire/employ a chauffeur.
The pilot needs to hold at least a PPL and to be appropriately qualified on the aircraft and for the type of flying (e.g. IR for IFR flight). So long as the aircraft is not used for hire or reward, in other words the people travelling on the aircraft do not pay for their travel whether with money or in kind, there is nothing requiring an AOC or similar. It's a private operation.
The aircraft insurer may have minimum requirements, e.g. CPL etc