Note: I have only read the first trilogy.
The third book revolves entirely around the fact that the First Foundation, knowing about the true existence of the second foundation, ruins the Seldon plan. Therefore, why did Seldon make it publicly known that the Second Foundation existed? How did this aid his plan for the First Foundation in any way?
My memory of the stories is that it wasn't commonly known that the Second Foundation existed.
There was a single mention of it in a report of one of the first planning meetings. That mention was basically "two Foundations at opposite ends of the galaxy." Nothing more, and there were no other public mentions of it at all.
Only Ebling Mis found that mention and followed it.
The equations the Second Foundation used depended on the mass of the people acting normally. They could predict the mass reactions fairly well.
As long as the mass of the galaxy's people behave normally, everything is fine.
Some few who know of (or believe in) the existence of the Second Foundation don't (mathematically) matter.
The issue is when the first foundation gleaned the nature of the second. That it existed was known from the first opening of time vault:
Terminus and its companion Foundationat the other end of the galaxy are the seeds of the Renascence...
At this point it was an independent site that had no impact on the first Foundation. There is no detail I can find on why this fact was shared, but we see there is detailed psychohistorical modelling of possible paths and events, so would assume that knowing they aren't the only Foundation helped to nudge them on to the correct path.
It became an issue and led to them trying to find and destroy it when they discovered that the second Foundation was directly interfering with the first.
I think the real reason is that when Asimov wrote the first Foundation book (or rather pieces that later become the book) he probably didn't think a 2nd Foundation would be any different from the first one and so there was no reason for keeping it secret in his mind.
When he then begun expanding Foundation universe and later merging it with his other works, only then it become "a problem" (or a "solution" for how to keep Foundation universe interesting). There are more examples where Asimov clearly wanted one thing in older books and then later decided to change it - like place of origin/nature of The Mule, source of radioactivity on the Earth or existence of non-human intelligent species in our galaxy.
