The strength needed doesn't depend on the size of the bow, but on the stiffness of the bow. The best metric, just as for modern non-compound bows, is the draw weight, which is the force needed to hold the bow at full draw. A wide range of draw weights were used, but we can broadly generalise:

• Hunting bows might have draw weights of 50lb, or possibly higher. This is similar to modern Olympic target bows, modern hunting recurves, etc. This is a high draw weight for a beginner, but a fit and strong experienced archer will manage this easily enough.

• Military bows, especially from 1000AD onwards, are often 100lb and up. The draw weight of the Mary Rose longbows is variously estimated; the estimate of their average draw weight is about 130lb. A surviving Mongol conquest-era bow is estimated at 166lb. Surviving Ottoman military bows average over 100lb.

For the same draw weights, the strength needed would be similar to that for modern recurve bows. The advantage of modern materials isn't that the bows are easier to draw, but that they're more efficient - less energy is lost to hysteresis in the materials (stretching of the bow string, not all of the energy going into bending the wood being returned as it un-bends) and lighter stronger materials. The Asian-style recurve-reflex composite bows can be harder to draw at the start of the draw, because they start with a higher tension in the string due to pre-stressing, but the bigger issue as far as strength goes is the force needed at the end of the draw (i.e., the draw weight).

For target shooting, and for hunting, a high enough draw weight to be able to shoot arrows with high velocity is important. High velocity gives a good flat trajectory, and makes it easier to hit the target. For hunting, the arrows also needs enough energy and momentum to kill/wound the animal. A relatively modest draw weight of around 50lb tends to be sufficient for this, which is why many modern and ancient hunting and target bows have such draw weights. It also appears that many early military bows might have had similar draw weights.

The 100lb and up military bows are an attempt to defeat armour. Against armour, very high arrow energy is important, and high draw weight and long draw lengths are the keys to higher energy. Sometimes, quite heavy arrows were used, which increases the efficiency of the bow, giving more energy (at the cost of range). Unless armour is common, there is little benefit from very high draw weights. When armour is common, and one wishes to defeat it, very high draw weights become essential. Modern target shooters and modern and ancient hunters generally don't/didn't shoot at armoured targets, so their draw weights are/were lower.

For high draw weight bows, it isn't just a matter of strength. Technique is essential! But this is the same for modern high draw weight bows.