For anyone other than Chris Townsend or Steve Vickers,
I think it is simpler just to ask about the forgetful and free functors
$\mathbf{Frm}\rightleftarrows\mathbf{Dcpo}$
between the categories of frames and directed complete partial orders.
Not altogether surprisingly, $\mathbf{Frm}$ is the category of algebras
for the monad over $\mathbf{Dcpo}$.

There is a general question
for any adjunction $\mathcal{C}\rightleftarrows\mathcal{D}$:
Suppose we replace $\mathcal C$ by the category of algebras
$\mathcal{C}'$ for the monad over $\mathcal D$ that is induced
by the adjunction,
then $\mathcal D$ by the category $\mathcal{D}'$ of coalgebras
for the comonad on $\mathcal{C}'$, and so on.

In fact, this stabilises with the algebras over the coalgebras
under the algebras.  Indeed, only two steps are needed if we
started with categories in which idempotents split. Steve Lack
first gave me the proof of this.

Now suppose that the base category $\mathbf{D}$, in Chris's
case $\mathbf{Dcpo}$, has finite products and
the monad has a strength, which it does in the case of $\mathbf{Frm}$.

Then the category $\mathcal{D}'$ of coalgebras under the algebras
also has finite products.
These are the coproducts of algebras over $\mathbf{Loc}$ in Chris's question.

As least so it says in some notes of mine called
underlyingset/universal possibly from March 2007.
I might be persuaded to re-upload these notes into my brain
to see whether the proof is correct.

The question of the esistence of coequalisers of algebras for a monad
(or equalisers in $\mathcal{D}'$ in my setting)
is a notoriously difficult one.