So suppose we didn't have historical examples to refer to ...
FYL is steering you straight by suggesting looking at the data sheets, particularly if you look at the numbers for max signal (often for 2 tubes; remeber to divide the value in half).
What's a worst case? How about a single tube's screen pulling 30mA (could be much more in certain cases)?
30mA * 1k = 30v, and 30v * .03A = 0.9w. Double that and you get a 2w resistor safely derated; of course, if your resistor is 470 ohms, you get half the voltage drop and 1/4 the power dissipated. Which tells ya that 1w resistors are fine (and often seen in old amps).
Shouldnt you also add a little extra in w. rating for safty from heat build up, since the screen resistor is located on the output socket?
You can, and the cost of a 3w resistor is not too much. Or you could decide that it will take a couple-decades to overheat, and that when it does the value will go up. So maybe you don't care so much.
Remember -If- that screen resistor gives up the ghost output tube goes with it ? then maybe the OT ?
Recall what function the screen provides: in a tetrode, pentode or beam power tube, the screen has more influence over plate current than the plate. As long as the screen voltage stays relatively high, the plate voltage can swing very high or very low with very little impact on actual plate current (this is good for making big output power).
If the screen resistor opens due to too much heat dissipated, then screen voltage should drop to zero. Plate current should be well below its normal idle value (though perhaps not quite zero). The amp makes little, if any, clean power but doesn't die, and there's no further damage to the amp.
For the screen resistor to damage the tube or OT, it would have to short to something:
1. Short to ground - screen voltage drops to 0v, tube current lower, possible power supply damage (burnt series resistor or maybe choke).
2. Short to plate - screen and plate see near-same voltage all the time; functionally equivalent to triode operation. Lower output power, no damage.
3. Short to filament - high voltage on filament likely to damage the tube, maybe (*very* unlikely) the PT filament winding. Screen resistor has no effect either way.
4. Short to control grid - high positive voltage on the control grid makes the tube pass huge current, can damage the cathode, control grid, screen and/or plate; if fixed-bias, the tube likely dies, may damage OT and/or power supply components.
So as a quick guesstimate, the screen resistor would need outside help to damage anything if it were to fail, and that outside help (likely an unintentional short) would probably kill anything that would likely die regardless of whether the screen resistor were present.