If you go through your own insurance, you pay the deductible up front, but after the claim is settled they will pursue the other person's insurance and get it back for you. It will not affect your "record" with your insurance company - they're going to find out about the accident whether you claim it with them or not, but you're not at fault, so it won't significantly affect your standing.
If you go directly through the at fault person's insurance, you will not have to put out your deductible, but you will have to wait for them to investigate. Sometimes this can be done quickly and other times it takes a while.