The recent preprint Determining Generic Point Configurations From Unlabeled Path or Loop Lengths by Gkioulekas, Gortler, Theran, and Zickler, treats a generalization of this question to higher dimensions, and to the situation where the list of distances provided may not correspond to the lengths of single edges, but may also be the lengths of arbitrary paths and loops on the graph.

Complementing the counterexample in Vladimir S Matveev's answer, the recent preprint Determining Generic Point Configurations From Unlabeled Path or Loop Lengths by Gkioulekas, Gortler, Theran, and Zickler, finds a positive answer under certain circumstances to a generalized question.

First, they treat all dimensions $d\geq2$, and second, the list of distances provided (the "measurement set") does not necessarily correspond to the distances between pairs of points, but may also be the lengths of arbitrary paths and loops on the complete graph on the point set.

The main result roughly states that provided: (1) the measurements come from a generic point set (thus ruling out examples like the one in Vladimir S Matveev's answer) and (2) the list of measurements "allows for trilateration" (meaning that there are enough measurements to inductively construct full-dimensional simplices), there is a unique point configuration (up to congruence) consistent with the measurements.

The recent preprint Determining Generic Point Configurations From Unlabeled Path or Loop Lengths by Gkioulekas, Gortler, Theran, and Zickler, treats a generalization of this question to higher dimensions.

The recent preprint Determining Generic Point Configurations From Unlabeled Path or Loop Lengths by Gkioulekas, Gortler, Theran, and Zickler, treats a generalization of this question to higher dimensions, and to the situation where the list of distances provided may not correspond to the lengths of single edges, but may also be the lengths of arbitrary paths and loops on the graph.