[Boromir88]

As far as why the Eldar gave the palantiri to the Faithful and not the King of Numenor:

These stones were gifts of the Eldar to Amandil, father of Elendil, for the comfort of the Faithful on Numenor in their dark days, when the Elves might not come no longer to that land under the shadow of Sauron.~The Silmarillion: Of the Rings of Power and the Third Age

This should be able to give a bit of a timeframe when Amandil received them, and they were probably the Faithful's most valuable weapon against Sauron and Ar-Pharazon. It does make sense too as far as how Amandil, Elendil, and his sons were able to coordinate the Faithful's exit from Numenor.

The palantiri were no doubt never matters of common use or common knowledge, even in Numenor.~Unfinished Tales: The Palantiri

The 7 given to Amandil were thus probably kept under tight secrecy and security, since the Eldar had gifted them to aid the Faithful. Not anything you'd want Sauron to get his hands on (as would be proven in the next Age ) and not anything you'd give to the Kings of Numenor who were turning unfriendly to the Eldar.

Edit:

Which brings up the question of how exactly do they work?~Snowdog

I'm not sure if you mean how they work as in how one uses them (Unfinished Tales goes into good details about that) or how they are actually able to view far away places, and be used for communication?

They can't transmit sound but they transmit thoughts, which is how they appear to most commonly be used for communication.

For the most part they revealed only things near to another kindred Stone, for the Stones each called to each; but those who possessed great strength of will and of mind might learn to direct their gaze whither they would.~The Silmarillion: Of the Rings of Power

For the most part, the Stones were connected to eachother, somehow both visually and mentally (but not audibly). So, Telchar comparing them to webcams is a good one. How they made this connection with eachother is anyone's guess.

If we go under the assumption that Feanor made them (a pretty safe assumption). Since, Gandalf alludes to this possibility in LOTR and in The Silm:

'and other crystals he made also, wherein things far away could be seen small but clear, as with the eyes of the eagles of Manwe.'~Of Feanor and the unchainging of Melkor

Is it too outlandish to say Feanor was able to capture the light of the trees and put them into the Silmarils, that he captured the Eyes of Manwe's eagles and encased them in crystals! Must be a lot of blind birds in Valinor.