No such thing as a private CofA under EASA. If said aircraft doesn't have an EASA CofA, it soon will have. No reason PPL training can't be done under an EASA Standard CofA. The bigger issue is insurance.
Recently looked into this for a group member who had a huge change of circumstances just before completing his PPL, right at the same time we bought our aircraft. He's now ready to get back in the saddle, and quite rightly he would like to do it in the aircraft of which he owns a share.