Two tracial von Neumann algebras are elementarily equivalent if they cannot be distinguished by first-order sentences or, more algebraically, if they have isomorphic ultrapowers. The same definition can be made for (countable, discrete) groups, and it is natural to wonder whether or not there is a connection between two groups being elementarily equivalent and their corresponding group von Neumann algebras being elementarily equivalent.  In the first part of the talk, I will give examples to show that, in general, there is no connection in either direction.  In the second part of the talk, I will introduce a strengthening of elementary equivalence, called back-and-forth equivalence (in the sense of computability theory) and show that back-and-forth equivalent groups have back-and-forth equivalent group von Neumann algebras.  I will also discuss how the same is true for the group measure space von Neumann algebra associated to the Bernoulli action of a group on an arbitrary tracial von Neumann algebra.  The latter half of the talk represents joint work with Matthew Harrison-Trainor.