As I understand, this question: a) pertains to modern Platonists' thinking, and b) has a false assumption behind it.
I'm going to take a chance and make an assumption of my own: if I succeed in explaining (b), then (a) becomes a moot point. If, however, you want to know (a) even if it's hypothetical -- you don't need to read any further.
The false assumption (mentioned earlier in the comment by polcott) is about abstract objects exiting in their own realm as the non-physical timeless forms. And in Plato days it was perfectly reasonable to envision this kind of magical cloud storage for concepts. In purely design terms was is way more efficient than the actual implementation. Unfortunately, we cannot ignore the fact that it needs magic to work, not anymore.
The recent advances in Artificial Intelligence accidentally shed the light on the most mysterious aspects of the human psyche -- emotions, intuition, consciousness, qualia -- things that we perceive as magical or supernatural. Now we have a model that explains away the magical stuff.¹
For the abstract objects specifically (and just as the rest of the concepts/ideas we develop), it means each of them is but a piece of information:
We can always think of it as a binary string -- because no matter how it is actually encoded, it can always be translated into the famous sequence of ones and zeros. And those things are very much real, existing as records on some physical media. In this particular case, that physical media is our brains. Every person that has acquired an idea of a square, has their very own copy.
And they are copies -- very often they are actually replicas, as it is the case with the square. Other times every person comes up with their own ideas, as Mary did after seeing red for the first time. And, given the (very observable) fact that we all share, and are parts of the same objective reality, we end up sharing many identical (like ) or similar (like ) or, at times, even predominantly wrong² ideas about it.
Incidentally, it also creates an illusion of every concept existing as a singleton that we all share.
¹ available upon request
² like 'cmon, what do you mean that wasn't funny?!