Good philosophical question.
As our personalities and minds are all in the brain, if we were to transplant a brain it technically wouldn’t be a brain transplantation but a body transplantation. Your personality, emotions and memories would now be in a new body.
Going at it from a practical point of view it would be an incredible task (impossible) to wire up all the nerves that you would have to cut in order to physically transplant the brain from a donor to a recipient!
That aside, memories are thought to be stored in the relative strength of the connections (synapses) between nerve cells so I suppose that following that (impossible) re-connection you would experience the world through another person’s brain. I think how long that would last would then depend on when you had the transplant as early on in life the brain is more plastic and can be more actively sculpted by experiences. All sensory inputs into the brain under go what is called a critical period of plasticity. If the transplant was performed after that time then there would be a greater chance of the host brain having a continued influence!
Going from science fiction to science fact: some groups are trialling transplantation of nerve cells into recipient brains to overcome intractable (drug-resistant) epilepsy and even a while back they trialled a similar approach for neurodegenerative conditions. In both these cases the injected cells would have to form new connections and so not retain any memory. I think this is a more likely scenario!