Yes, privately built drones can be used, and depending on their weight, operated in the ’open’ category or the ’specific’ category. You, as the drone operator, need to fulfil all the requirements of the Regulation, and in the ’open’ category, you can only operate under subcategory:
A1 when the drone’s maximum take-off weight (MTOM) including its payload is less than 250 g and the maximum speed is less than 19 m/s; or in
A3 when the drone’s MTOM including its payload is less than 25 kg.
Regulatory reference: UAS.OPEN.020 (5) (a) and UASOPEAN.040 (4)(a) Annex part A of EU regulation 2019/947.