Actually much of the time the contract is signed only by the Contracting Officer. In order to establish a mutually binding legal contract between the buyer and seller, we need to have an "offer and acceptance". Typically, when the contractor submits their bid or proposal in response to the Government's solicitation, the bid or proposal (with SF33 signed by offeror in block 17) is the contractor's offer. When the Contracting Officer signs the contract (in block 27) , the signed contract is the Government's acceptance of the contractor's offer. At the time of contract award, the contractor isn't required to sign the contract. However, things get a bit more complicated if a term and condition in the contract is different than what's in the proposal (probably as a result of discussions (negotiations)). When discussions are held under FAR Part 15 competitive procedures, the offerors are given the opportunity at the conclusion of discussions to submit final proposal revisions (FPRs). As long as the apparent low offeror's FPR reflects what was negotiated between the parties during the discussions phase of the source selection, the contracting officer signs the SF33 on the contract and no contractor signature is required at time of contract award.
A sole source negotiated procurement under FAR Part 15 throws a wrench into things though. In this case, it's always a good idea at the conclusion of negotiations to have the contractor submit a revised proposal which reflects the most current and up-to-date terms and conditions. The Contracting Officer normally awards on the SF26 in which case the contractor would be required to sign the contract too. Most Contracting Officers prefer to use the SF26 for any negotiated sole source procurement regardless of whether or not we were able to get an updated proposal from the contractor in order to leave no doubt that there is mutual agreement between the parties.
One exception to "offer and acceptance" as described above, is when a purchase order is issued under FAR Part 13 Simplified Acquisition Procedures (SAP). Here, in accordance with FAR 13.004, the vendor's quote is NOT considered an offer. The offer happens when the Contracting Officer issues the purchase order to the vendor. Acceptance occurs when the contractor starts performance or signs the purchase order whichever happens first. Another situation that's a little unique is a letter contract (See FAR 16.603 and DFARS 216.603). Here, in accordance with FAR 16.603-4(b)(1), the Contracting Officer is required to include FAR Clause 52.216-23, Execution and Commencement of Work, except that this clause may be omitted from letter contracts awarded on the SF 26. This clause requires the contractor to indicate acceptance by signing the contract and returning it to the Contracting Officer.