Just so this has an answer, let me expand upon what I said in the comments slightly.

So, let us begin with the following trivial observation:

**Observation:** If $f:E\to E'$ is an isogeny of elliptic curves over $k$, with $\mathrm{char}(k)=p>0$, then $f$ and $\widehat{f}$ are separable if and only if $p\nmid\deg(f)$.

Indeed,this follows immediately from the follow facts: in a tower $E/K/F$ the extension $E/F$ is separable if and only if $E/K$ and $E/F$ is separable, $[n]$ is separable if and only if $p\nmid n$, and $f\circ\widehat{f}=[n]$.

Thus, your question comes down to whether or not there even exist separable isogenies $f:E\to E'$ with $p\mid \deg(f)$. The answer is yes if and only if $E$ is ordinary.

To see this most clearly, note that every isogeny $f:E\to E'$ is of the form $E\to E/K$ (quotient map) for $K\subseteq E$ a finite subgroup (scheme) and that $\ker f=K$ so that $\deg(f)=|K|$. Moreover, $f$ is separable if and only if it's étale (separable=generically étale=étale for group schemes since one can translate the generic étaleness everywhere) if and only if $K$ is étale (over $\text{Spec}(k)$).

Now, we may as well assume that $\deg(f)=p^r$ for some $r$ (since this is the only case of real interest to us) in which case we see that $K\subseteq E[p^r]$. Thus, your whole question comes down to whether or not $E[p^r]$ has an étale subgroup scheme. Moreover, it's easy to deduce from the fact that $E[p^r]$ is an iterative extension of $E[p]$ that this is true for some $r$ if and only if it's true for $r=1$. Thus, the existence of an isogeny $f$ which is separable but for which $\widehat{f}$ is inseparable is equivalent to the question of whether or not $E[p]$ has an étale subgroup scheme.

But, this is precisely equivalent to whether or not $E$ is ordinary or supersingular. Namely, depending on your definitions, $E$ is defined to be ordinary if $E[p]$ has a non-trivial étale subgroup (and supersingular otherwise). If you, instead, use the definition that $E$ is ordinary if and only if $|E[p](\overline{k})|>1$ note merely that a finite scheme $X/k$ is étale if and only if $\#(X_{\overline{k}})=\dim(\mathcal{O}_X(X))$. Using this it's easy to see that this definition of ordinary agrees with the one I said above.

To summarize, the only interesting isogenies in this question are of the form $f:E\to E'$ with $\deg(f)=p^r$ for some $r\geqslant 1$, and are of the form $E/\to E/K$ (quotient map). Moreover, this $f$ is separable with inseparable dual if and only if $K$ is an étale subgroup scheme of $E[p^r]$, which exist if and only if $E$ is ordinary.