Beliefs about equilibrium paths with overlapping strategies
In signaling games, beliefs as well as strategies constitute an equilibrium. In other words, beliefs are a part of the equilibrium object. By deciding which equilibrium to focus on, you also (partially) determines the beliefs consistent with that equilibrium. The key restriction is that beliefs be derived from Bayes' rule using sender's strategy and the prior. 

Consider the following game, where P1 is the sender and the common prior is Type I realizes with probability $p$. 

Separating equilibrium



In a separating equilibrium, different types choose different actions (Type I chooses $A$ and Type II chooses $B$). Then, upon seeing action $A$, P2 (the receiver) updates her belief using Bayes' rule:
$$
\mu(\text{Type I}|A)=\frac{\Pr(A|\text{Type I})p}{\Pr(A|\text{Type I})p+\Pr(A|\text{Type II})(1-p)}=\frac{(1)p}{(1)p+(0)(1-p)}=1
$$
Therefore P2's conditional belief at node $x$ is $1$. Similarly for her belief at the information set $\{x',y'\}$

Pooling equilibrium



Now, consider a pooling equilibrium, where both types choose $A$. Then, upon seeing action $A$, P2's belief is calculated as 
$$
\mu(\text{Type I}|A)=\frac{\Pr(A|\text{Type I})p}{\Pr(A|\text{Type I})p+\Pr(A|\text{Type II})(1-p)}=\frac{(1)p}{(1)p+(1)(1-p)}=p
$$
which is the same as her prior. This makes sense because P1's action provides no additional information in a pooling equilibrium. (P2's off-equilibrium belief can be arbitrary, unless you want to apply refinements such as consistency of the sequential equilibrium).



The bottom line is: you cannot talk about an equilibrium separately from the beliefs. By talking about a particular equilibrium, you also implicitly restrict the set of beliefs consistent with the said equilibrium.