Do tanker aircraft distribute their own fuel? Do tanker aircraft, (such as the KC-135, KC-10, S-3B, etc), distribute fuel from their own fuel tanks, or is it strictly from a separate cargo tank?

Obviously, some cargo tanks are going to be involved, as an aircraft carrying only the fuel in it's wing tanks isn't going to be a whole lot, and some of these aircraft are known to fuel some very thirsty birds, such as B-52's, etc. 

Sub question: What proportion volume of the aircraft's fuselage is taken up by the addidional fuel. Do they just pack it as full as they can with fuel? Seems like this would very quickly add to weight, and you would struggle getting such a fuel laden ass bird off the ground.
Both the KC-135 and KC-10 can burn the fuel it carries on board for refueling (the notable exception was the KC-135Q, which carried separate tanks of JP-7 for the SR-71 Blackbird).



Both indeed do have significant fuselage tanks - the KC-10 is notable for being 'dual-mission' capable, meaning it can carry both fuel and cargo, and the KC-10 is also air-refuelable (so a KC-135 can offload extra gas into a KC-10 when their crew rest is up, allowing the KC-10 to stay on station and refuel other planes longer). The maximum fuel load of a KC-10 is 356,000 pounds. This is over 7 full loads of gas for a B-52 (B-52G and H could carry 308,000lbs of fuel, so 365,000 capacity of KC-10 would potentially fully fuel a B-52 one time, with enough fuel left to get the KC-10 home. Of course the B-52 would never be completely empty in the air so the KC-10 would not need to offload the full 308,000lbs), or 30 full loads for an F-16 with external tanks, and the maximum fuel load of a KC-135 is 200,000 pounds, or over 4 full loads of gas for a B-52.

As for how much of the fuselage is taken up by fuel tanks, you can look at cutaways of the KC-135 and KC-10 (unfortunately the captions for the KC-10 are in a foreign language, as I could not find a high-enough resolution one in English). In both cases, the fuel tanks are in the lower half of the fuselage - the reason the KC-10 has a cargo mission as well is because it is a larger aircraft and has a much higher gross weight. The KC-135's cargo space is much smaller, and as such is not used for transporting large items. This technical document for the KC-10 also gives volume of each tank (called Fuel Cells):


Forward Fuselage - 8,250 gallons in 3 sections, 6' deep
Aft Fuselage - 9,870 gallons in 4 sections, 6' deep


You are correct that it is a struggle getting them airborne when fully loaded - several of the KC-135 pilots I have talked to have taken until the very end of 10,000 foot plus runways to get airborne, and one of the KC-10 pilots I know said they would occasionally take off with less than a full load then take on fuel from a KC-135 to make it easier.

Page 3-45 of this technical manual has a chart for calculating minimum runway for a particular speed - for example, a sea level takeoff at 30 degrees (Celsius) at Max takeoff weight would require about a 13,500 foot runway.