The Pen & Paper role-playing game Eclipse Phase deals heavily with these sorts of things. In that setting its quite common to spawn low level copies of your own mind to perform various tasks and then reintegrate them later on. The acceptability of this varies depending on where you are. Some societies see it as normal and accepted, while some think its queer and even obscene.
Generally a low-fidelity copy, one smart enough to perform basic things like managing your appointments, is fine but high-resolution copies capable of independent action and thought are restricted or even illegal.
You might spin off a copy of your mind to attend a work meeting and then integrate those memories later, while another copy of yourself gets the shopping done via online catalog, all while the 'real' you is sitting on the train off to visit your friend.
Under this fictional setting those copies are never granted independent rights but a copy of your mind is respected as if it were wholly you, and you are responsible for any actions taken.
And yes, out-of-control copies are a common plot point. In some societies, such copies are fitted with the software equivalent of a kill switch to remotely deactivate them in case they start causing trouble. As you can imagine, the morality of all this causes a lot of problems in the game and several factions have banned the practice outright.
This is also a game where you can simply download your mind into a comms network and transmit it to another space station where you're loaded into a loaner body, rather than travel there the slow way.