Employment Law Questions? Ask an Employment Lawyer.
Generally yes, for a contract to be binding, there needs to be executions of that agreement by both parties in most cases. They have your signed copy and they should have provided you with a signed copy from them.
Now, just because you have an unsigned copy that does not mean that they didn't sign it or didn't retain a signed copy for themselves. They could have signed a separate copy of the agreement and that would be fine.
Also, if the employer engages in the agreement terms, it would be reasonable to assume (legally) that they intend to be bound by the terms of that agreement even with having provided you a signed copy. You provided them a signed copy, so they can show that and hold you to the terms of the agreement. They can also conduct their part of the agreement, essentially binding them through their actions.
You can certainly just call them and ask for a countersigned copy.