It's semi-public. A player's set of bills must be visible, but they are not obligated to tell you how much money they have.
This comes from the 2009 Monopoly World Championship rules (emphasis is mine):
Your money must be in view of the banker, judge and other players during play. You can leave it in a pile and even turn it face down, so it’s not clear exactly how much you have, but it must all be on the table – you can’t hide a note in your pocket and put it back into play later.
• You can briefly count your money out of sight of other players (e.g. under the table) before putting it back in full view of others.
• You do not have to tell anyone exactly how much money you have, even if asked.
The best analogue I can think of is points in Small World. You can get a rough approximation from looking at the size of their pile of money, but they can use creative denominations to be misleading one way or another, and unless you've been keeping track, you won't know the exact number.
(I Thanks to @L.ScottJohnson, this is corroborated by Hasbro Customer Care:
Your money must be in view of the banker, judge and other players during play. You can leave it in a pile and even turn it face down, so it's not clear exactly how much you have, but it must all be on the table – you can't hide a note in your pocket and put it back into play later. You do not have to tell others how much money you have. They can guess based on how big your pile is. -- Rochonda, Global Consumer Care Representative.
Personal note: I was fairly surprised by this answer. It's at odds with how I've played the game, but it's the most authoritative source I can find.)