of course, one should expect that the concept of ring-valued points is not well-behaved for locally ringed spaces (LRS). I want to see examples for this.

so consider $LRS \to Set^{Ring}, X \mapsto X(-)=Hom(Spec - , X)$. if $A$ is a local ring, whose maximal ideal is principal, and $\hat{A}$ its completion, and we regard local rings as locally ringed spaces whose underlying set is just one point, then $A \to \hat{A}$ induces a bijection $Hom(Spec R,A) \to Hom(Spec R,\hat{A})$ (I'll add the proof if you want). this shows that the functor is not full. but how can we see that it is not faithful?

For example, for local rings $A$, we have

$Hom_{LRS}(Spec R,A)=\{\phi \in Hom_{Ring}(A,R) : \phi(\mathfrak{m}_A) \subseteq rad(R)\}$.

If $f,g$ are local homomorphisms inducing the same maps $Hom_{LRS}(Spec -,B) \to Hom_{LRS}(Spec -,A)$, it seems that they don't have to be identical ...