My understanding is that both miners and nodes are considered full nodes and that the nodes perform the consensus, but the miners add blocks to the blockchain and for this extra work they get rewarded in tokens. In the case of a private blockchain where there are no tokens being offered do you still need miners?
The reward will not be a reward but just a way to generate the coins. This is one of the major reason why these "Private"
Blockchains are doomed to fail. In "Public" Blockchains the people know the supply are fixed and it cannot be changed without
the consensus of the majority of the nodes. {hosted by the general public} In "Private" Blockchains these decisions are made
by a central authority, like it is done by reserve banks now. {printing money like toilet paper} Now it can be done, by a push
of a few buttons. {scary stuff}