The barium that might have leached will be precipitated by the carbon dioxide from the air into barium carbonate.

Any form of soluble barium left in the environment will end soon as either barium sulfate or barium carbonate, which are insoluble, thus non-poisonous.

Barium is one of the least dangerous metals with high atomic number. You have to do something really stupid and deliberate with it in order to be able to poison anyone.

Most white paints used nowadays contain titanium white.

Nevertheless, many white plastics contain other white pigments instead of titanium white. Titanium white itself is inert, but when it is exposed to the sun light it can catalyze the destruction of organic substances like plastics or paint binders that have not been selected to be resistant to this.

So when using titanium white outdoors, the white paint must be specified as UV resistant.

I have not heard about paper that uses titanium white.

That might be due to the higher cost of titanium white, which might be too much compared to the cost of paper.