There were two copies of the register, one being a copy in the Registrar's handwriting. One copy was sent to Edinburgh and one was retained by the Registrar. Sometimes it was the one with the original signatures that went to Edinburgh, and sometimes it was the copy. It is the Edinburgh copies that were digitised for Scotland'a People.
The local registers retained by the Registrar were sometimes stored in a central repository for the county. For example the pre-1900 retained registers for the whole of the county of Moray were, and I think still are, in the Registrar's office in Elgin, and the later ones were still held by the local Registrars. Likewise the pre-1900 Banffshire ones were, and may still be, in Banff, and the later ones were in the various local Registrars' offices. The pre-1900 ones for Aberdeenshire and Kincardineshire were all in Aberdeen. However I have not visited any of these places for many years because they started charging silly money for access to them in the county and local Registrars' offices. There was talk some years ago of destroying the retained copies of the registers, and I have not heard the outcome of that. Nor do I know where the retained registers for Glasgow were or are kept.
No official certificates were retained by the churches, though the churches probably do keep some sort of records for their own purposes. You have to remember that it was very unusual for a wedding ceremony to be held in a church building, so there was no opportunity for a ceremonial retreat to the vestry to sign the register. Most weddings were held in the bride's parents' home. If she had no parents, or was getting married a long way from home, the next most common place was in the manse (the minister's house). Only towards the end of the 19th century did weddings in hotels, restaurants and halls come into fashion, and church weddings didn't really become the norm until well into the 20th century. I was the (only) bridesmaid at a friend's wedding in her parents' home in about 1978 or 1979.