Out of universe, it is worth mentioning that the several Foundation instances were not a single work, but created & sold separately to SF magazines (so, the first Foundation book was actually five different stories in the same universe). Foundation and Empire, likewise, were two different long stories, (The general & The mule).
So, it is probably that it was just a cliffhanger for a future work set in the war of the first versus the second foundation. At that point, the arc still had not been spoiled by the Mentallics "idea", and little had been explained about the Second Foundation (so Asimov still had a great degree of freedom).
In universe, the First Foundation had been attacked by the entity that still was, with difference, the biggest military power in the galaxy. And not only the Foundation did repel the invader, it got a new province and captured ships. In the eyes of the Foundation "physics" (who would see the history as a matter of who has more money/technology/ships/terrain) it was clear that if the Empire had failed that time, it would fail again if he tried (as the Foundation would be progressively stronger, and the Empire weaker), and that the Second Galactic Empire was almost unavoidable.
Of course, they thought that at some time they would find the Second Foundation. And they thought that it would have the same nature of the First Foundation (expansionist, imperialist even). Perhaps they even thought that the "securities" that the Second Foundation gave to the Plan were that, in case the First had failed, the Second would have followed the path that was assigned to the First.
In any rate, they expected that when the Second Foundation was found, it would be a competitor, not an ally (*1). Even if a deal could be struck with the Second Foundation, it would imply that the First Foundation ruling class would have had to share power with the ruling class of the Second Foundation. And given how little that ruling class liked the idea of sharing any power (*2), they would have to assume that the same would happen to the Second Foundation ruling class.
*1: At that time, probably there would be no other kingdom or alliance capable of being a credible "common enemy" to both of them.
*2: Remember that at the time the First Foundation had an oligarchic government, and that relying in independent traders was almost a "last resource" option for them.