Looking for a replacement for a Leviton model 770W 3 rocker 4 function switch

SOURCE: Leviton Single Pole Switch with Pilot Light Model 277W Box

Depends on whether you want the pilot to glow when the circuit is live (the main light is on) or to glow when the circuit is off. (The switch can be used either way, and I have used this switch in both configurations at different places.)

The wiring diagram is on the inside of the box.

In indicates that if you want the pilot light to glow when the circuit is off, then you only need two wires (the live and the return, usually colored black and white, respectively). However, you also need a ground. The instructions say that you need the ground for safety (and probably legal code) reasons, but I suspect that, without the ground, the pilot light won't work in this configuration.

However, if you want the pilot light to glow when the circuit is on, you probably need a three wire configuration (plus the ground).

Hope this helps.

SOURCE: not sure how to wire this switch

Switch will have a commom terminal that the hot wire connects to.
The other three terminals are for the devices -- light, fan and other device.

The white wires should be the grounded wire returns for the circuits unless they are identified by color tape to be hot line side of power.

Be careful of the grounded side of circuit, can shock or kill, treat all wires as being energised. If you are not sure, use a electrical tester,
or get help.

SOURCE: A Bryant 3 way independent rocker switch with red,

Leviton triple switch
Nutone P66V

Nutone looks like an ordinary three-switch combo being marketed as a bathroom 'fan switch.'
The screws on the device would tell the story.

Look at the Leviton screws:
The switch has dark-colored screw on one side >> the hot from breaker always connects to dark screws.
Other side of switch has 3 silver-colored screws, but these can be brass colored screws > the wires to fan light and heater connect to each of these screws.

Most likely the Nutone switch is identical to the Leviton
I say this because the descriptor on Amazon describes it as a 'fan' switch.
It's being marketed as a bathroom fan-light combo switch >>> so it looks like an ordinary three-combo switch

SOURCE: I replaced my old 3way

http://www.leviton.com/OA_HTML/ibeCCtpItmDspRte.jsp?item=4044§ion=10103&minisite=10025
The Leviton page for Leviton 5613 does not have a troubleshoot.
Manual shows simple wiring diagram.
3-way switches will not operate lights if switches are wired incorrectly.
Check your ground wire is connected.
Leviton 5613 series is illuminated.
Check that both switches have 5613 number.
Swap switches and see if problem is inside switch.

SOURCE: Do I connect the black and red wires coming from

http://www.leviton.com/OA_HTML/ibcGetAttachment.jsp?cItemId=sPOj1y4-FvozD26dzcf2mQ&label=IBE&appName=IBE&minisite=10026
Here is the wiring diagram.
It looks like you connect the black & red wires on the sides as you state.
The black coming from one 3-way connects to the top terminal on one side, the red coming from the same 3-way connects to the bottom terminal on the same side.
The black going to the other 3-way connects to the top terminal on the other side and the red going to the other 3-way connects to the bottom remaining terminal.