Just take the resulting points and past them into EuclideanDistance... but remove the outer brackets or use Apply as in
EuclideanDistance @@ {{1.384, 1.878}, {1.896, 2.33}}
one also could certainly create a tool to do this using things like this (it gives the distance between the last two points created--create new points by alt-clicking on windows or command-clicking on OSX):
Manipulate[
Column[{
Plot[x^2, {x, -1, 4}],
If[Length[u] > 1, EuclideanDistance @@ Take[u, -2], 0]}
], Button["reset", u = {}],
{{u, {}}, Locator, LocatorAutoCreate -> True}]
