Title search through attorney?

Hello…since attorney’s are used for real estate transactions here in the Northeast, is it preferable to pay for the title search through my attorney BEFORE trying to assign the contract, or find the buyer and let them do it through THEIR attorney? I know you like to show your prospective buyer clear title, but logistically it is just simpler to assign the contract and let the buyer’s attorney do most of the paperwork. The assignment fee can be due at clear title. Any thoughts?