Is it possible to only mine your own incoming transactions? So you can confirm a Bitcoin transaction (only locally).
This might only be an issue with the terminology, but when a transaction is "mined", it is included in a block. You could do that of course, if you were contributing blocks to the Bitcoin network, but the computing power of the network has grown to levels where it is infeasible to mine at home.
What you can do, is to verify the validity of an incoming transaction. Your Bitcoin wallet software does this automatically when it receives the transaction. It checks that the inputs are available for spending, and the signature matches the source of the funds. However, until a transaction is part of a block in the longest valid chain, it is unconfirmed and may yet be replaced by a competing transaction spending the same funds. Therefore, you cannot "confirm a transaction locally".