We introduce nondeterministic streaming string transducers (NSSTs) { a new computational model that can implement MSO-definable relations between strings. An NSST makes a single left-to-right pass on the input string and uses a finite set of string variables to compute the output. In each step, it reads one input symbol, and updates its string variables in parallel with a copyless assignment.We show that the expressive power of NSST coincides with that of nondeterministic MSO-definable transductions. Further, we identify the class of functional NSST; these allow nondeterministic transitions, but for every successful run on a given input generates the same output string. We show that deciding functionality of an arbitrary NSST is decidable with PSPACE complexity, while the equivalence problem for functional NSST is PSPACE-complete. We also show that checking if the set of outputs of an NSST is contained within the set of outputs of a finite number of DSSTs is decidable in PSPACE.