How do I get private bookmark from web Google Maps to Android Google Maps? I have few private placemarks stored to "My Places" on Google Maps web version using same Google account than my Android Google Maps uses.

How do I get those places to the Android?
There are two ways I think you can do this, in the Android app, click on the layer selection and go to the My Maps layer, where you should see your private maps.

If you want to access your bookmarked address, from the main Maps screen, click the menu button and choose the bookmarks option. (not sure about bookmarks, I have a French UI, so it shows as 'favoris')