Always.

In the case of algebraic theories, the proof is simple.  Let $F : Set \to V$ be the free functor, where $V$ is some algebraic theory.  Then if $X$ is a set and $U$ a $V$-algebra,

<p>
$$
\operatorname{Hom}_{V}(F(X),U) \cong \operatorname{Hom}_{\operatorname{Set}}(X,|U|)
$$
</p>

The right-hand side is a $V$-algebra, naturally in both $X$ and $U$.  Therefore $F(X)$ is a co-$V$-algebra object in $V$, which is more structured than just being a $V$-algebra.  Furthermore, if $X$ is a monoid then $F(X)$ becomes a Tall-Wraith $V$-monoid (obligatory n-lab reference: [Tall-Wraith monoids](http://ncatlab.org/nlab/show/Tall-Wraith+monoid)).

(The details of this particular argument are a part of a [paper by Sarah Whitehouse and myself on Tall-Wraith monoids](http://www.math.ntnu.no/~stacey/Research/Preprints/free.html).)