Only about a dozen countries can launch satellites. Russia, China and the United States are the major powers when it comes to putting stuff in orbit. New technologies, like more available rockets, and smaller "micro-satellites" (under a hundred pounds, but with the capabilities of much larger birds of a decade ago), mean more countries are going to be putting satellites up. In the next decade, many countries will have dozens of satellites, mostly micro-sats, in orbit. American dominance will be remembered as a brief period (a decade or so), after the Soviet Union disappeared. America will continue to have the edge in satellite capability (the best resolution, and abilities to detect many more things), but they are already a minority in terms of all the satellites up there.
It's getting crowded up there. Currently, over 800 satellites are in orbit, owned or operated by 41 different countries. Most nations are only concerned with communications in space, and 85 percent of the countries that operate satellites, operate communications birds. A growing use of space is surveillance (mostly photo sats), and about a third of participating nations have one or more of these. About a third of the satellites up there are military, or dual military/civilian use. Most of these are operated by the United States. The Soviet satellite fleet, which was often larger than the American one, starved to death during the 1990s. Lack of money, in the wake of the collapse of the Soviet Union in 1991, led to few replacements going up. Year by year, the Soviet satellites wore out and went silent. Only now is Russia beginning to send more satellites up. But the Soviet era fleet is history. Russia will never pay for that many satellites.