Do both the landlord and tenant have to sign a lease agreement?

Yes.

Normally yes, there can be at times assumed, when one or both parties act and rely on the lease as a valid agreement. For example if the landlord is not present but his agent agrees to the lease, and takes your money and allows you to move in. You can make a case of assumed validity of the lease.

Also if you sign the lease, and they give you a copy, it can be believed that the landlord has signed it ( and if you try to take them to court they will have by the time you get to court0