Because Mindfulness and other types of Focused Attention Training derive from religious practices, many people may feel that they are private or non-public activities.

There is some merit in this line of thought. The practice, any that I've seen, is private in that you are focusing inward to improve attention and enhance awareness.

But why really does this seem private?

There is nothing really private or revealing about it. Your thoughts aren't different when meditating than at some other time. Nor are they required to be shared with others. Nor are they shared with others.

What differs, really, is posture. The eyes are closed. You're sitting there. It's prayer-like in that way, even if what you're doing isn't religious. You're just paying attention to the mind. This resembles sleeping or praying and those are considered to be private activities in the West.

I guess sleeping is private in every culture. And it is the closed eyes, probably more than anything, that make it seem private to us.

Mostly it's different. If you aren't with a group of people, you're on your own and that can feel uncomfortable. Better to do that thing you're a little unsure of in private.

But ask yourself these questions:

What seems more personal to you, reading a book or meditating? Which would you be more likely to do in public? Which is actually the more private activity? Why?

Mindfulness practices are maintenance and improvements for the machinery of the mind. If we think of them this way then there shouldn't be anything more personal or private about them than any other activity you engage in for self-improvement.

Does that mean you should go out an meditate in public? I guess that's a personal preference and probably relates to your goals and why you meditate. Some people do it.

For the rest of us, just something to consider.