One thing that is usually seen as something “in the mind”, but which I think is public are possibilities. For example, if I see a cup on the table, I can be aware of the possibility for the cup to be taken and put in some other place. One might be inclined to think of this as something going on in the mind, something private.
I can think of two reasons why they are assumed as something “in the mind”.
1. While things are usually clearly seen by everyone in specific place, this is not the case with possibilities. If you get two people to same place, and have them look into the same direction, (e.g. a chess table), they will easily report all the pieces, but one of them (let’s say a chess master) will report some possibilities (e.g. about how white can win) that the other person can’t report. So, those possibilities are something which is in the master’s mind, but not in the amateur’s mind.
This I think doesn’t mean that the possibilities are in the mind, or that they are private. If two chess masters look at the same table, they might see the same possibilities – they can see what moves can make white win. They can’t see the same possibility for the white to win, if there are in fact two instances of possibilities in each of their minds.
Further the chess game is a special case, because the possibilities there are complicated one, and hard to see. In more normal cases like that of a glass on a table, the possibility to move the glass should be obvious to everyone. That some things are harder to see than others, and that some people can see some things, while other can not, isn’t argument that those things aren’t in the world.
2. We might think there is a possibility even when there isn’t such a possibility. I might think that the cup can be moved, when really the cup is glued to the table. So, the possibility is something in the mind.
This again, doesn’t follow. That we can mistake a leaf carried by the wind for a butterfly, doesn’t mean that there are no butterflies in the world.
Sometimes I become aware of the possibility only after watching some other person doing it. Kids see grown up person doing something and then they become aware of the possibility to do that, so they soon start doing those things. Possibility to talk, possibility to type on computer keys, possibility to make grimaces, possibility to move things, open containers, to put things into containers, kick ball with a foot, put water in a glass, and so on.
The objects and possibilities are seen in the world, and as there for others to see. If most of the things including the possibilities are in the world, we know of other minds not through deducing some hidden and private realm in people’s heads from “their physical behavior”, but is the natural starting point before any theorizing – the things we see are there open to be seen by everyone that can see.
The problem doesn’t appear for kids, not because they do have some theories which relate the behavior to a Cartesian souls, but because they are not trying to limit the world to its physical aspect, which would then make them wonder why there are such things as possibilities, colors, sounds, etc…