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, ie the next step is $\mathcal{D}''\cong\mathcal{D}'$.
Indeed, only two steps are needed ($\mathcal{C}''\cong\mathcal{C}'$)
if we started with categories in which idempotents split.
Steve Lack first gave me the proof of this.

Now suppose in addition that the base category $\mathcal{D}$, 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.
The proof is quite complicated and works with strong monads in the abstract
rather than frames, though it was motivated by exactly this question.
I might be persuaded to re-upload these notes into my brain
to see whether the proof is correct.

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