For an excellent lesson on this issue, see a classic article from Harry Browne:

The ninth and tenth amendments were included to make absolutely sure there was no misunderstanding about the limited powers the Constitution grants to the federal government.

Amendment IX:

The enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people.

Amendment X:

The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.

Now, where’s the right to privacy?

It is clearly in those two amendments.

The government has no power to tell people what to do except in areas specifically authorized in the Constitution.

That means it has no right to tell people whether or not they can engage in homosexual acts; no right to invade our privacy; no right to manage our health-care system; no right to tell us what a marriage is; no right to run our lives; no right to do anything that wasn’t specifically authorized in the Constitution.

(read more)

It’s pretty straightforward.  There is a right to privacy.  Why? Because the government isn’t specifically given the power to violate your privacy.

That’s what the 10th Amendment is all about – government is strictly limited to doing those activities which are specifically authorized to it by the Constitution.

Everything else is left to “the States, respectively, or to the People.“