Insulation has been installed on my bedroom ceiling with the white foam side showing and the shiny side against the ceiling. Does the exposed foam side pose any dangers?
The foam just sitting there is not a problem normally. Any residual chemicals from manufacturing will dissipate fairly quickly and the foam itself is quite stable.
The problem is fire. Many foams both burn easily and emit toxic fumes when they burn. Almost anything that burns will be toxic if you breathe in too much carbon dioxide and not enough oxygen, but foam can be extremely dangerous.
The usual solution is to have a fire barrier between you and the foam. Typical in the US is drywall. Drywall is basically ground up stone (gypsum) between layers of paper. It is quite good at slowing down a fire. And even wood can do fairly well - we think of putting wood in a fireplace but it doesn't actually catch fire all that fast. Bare foam? Doesn't matter shiny side or not, it is considered quite dangerous in a house fire.