No. Most domain name ownership transfers can be completed at the domain's current registrar. After the Buyer gains control and releases the escrowed funds, the Buyer can transfer the domain to a registrar of their choice. However, with the Seller's agreement, the domain can be transferred to the Buyer's receiving registrar before funds are released.
The Buyer should reject the domain on the Escrow.com site if complete control cannot be determined. Escrow.com also suggests using an ICANN accredited registrar.