You are going to need an escrow or settlement company to act as an “independent third party” between you and the seller. Without having a third party involved, how do you know that when you fork over the money, you are going to get the deed? This is the type of service provided by escrow and settlement. They will hold your deposit and coordinate much of the activity that goes on during the escrow period.
Since this third party is very important to both you and the seller and both of you will pay fees to this company, it is important to agree on which service to use. Therefore, your choice should be part of the offer. Since you do not buy a home every other week or so, you are probably unfamiliar with companies that provide this service. Your agent will make a recommendation. You have the authority to accept this recommendation and include it in your offer, or make your own choice.
Keep in mind that the seller will also have a preference and this may be a point of negotiation in a counter-offer. It has become customary that one side will choose the escrow/settlement agent and one side chooses the title insurance company. Even so, everything in real estate is negotiable.