Do tanker aircraft distribute their own fuel?
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.