Let $X$ be an algebraic space locally of finite presentation, and let $\tilde{X}$ denote the restriction of $X$ (as a functor on schemes) to the category of complete local rings. Is it true that the mapping $X \mapsto \tilde{X}$ (of algebraic spaces to functors on complete local rings) is a fully faithful functor?

I.e. can we uniquely determine a morphism $f : X \to Y$ of algebraic spaces locally of finite presentation simply by specifying its values on complete local rings?