There are several relevant factors here. For one thing, if the paper is to bend at all, it is largely constrained by the surface against which it is pinned; it cannot turn backward to penetrate the wall! So if it is going to bend at all, it must bend forward, or pretty nearly anyway.
Secondly, the wall is likely on the one hand to be donating moisture, and on the other hand discouraging the evaporation of moisture, so that the paper on the wall side might be damper than on the air side. Thirdly, there is less circulation of air between the wall and the paper than on the open surface of the paper. Commonly this too will favour drier paper on the open face. Possibly you will have noticed how the upper slice of bread in neglected sandwiches curls up as it dries. That effect is obvious and dramatic; the innards of the sandwich are moister and accordingly they discourage shrinkage. Slightly less obviously you also will find that if you remove such a slice from the sandwich and place it in a symmetrical position, or even one that favours desiccation of the most side, it is too late to reverse the effect. Some of the material has set permanently into that curved position and cannot reverse. (By way of analogy, anyone with experience of barbecues will know how important it is to turn meat before it commits to a given direction of curling as its fibres contract in the heat.)
The lamp near the paper is likely to dry the hanging sheet too fast to encourage much curling, but a more important if it might be that it also has dried the surface of the wall or the noticeboard sufficiently to cause it to absorb water vapour actively when it is available and sheltered from the lamp. Accordingly the inner face of the paper would not try more slowly than the outer face and once the paper is properly dried and in equilibrium with the air on one side and the wall on the other, it is likely to remain fairly flat.
My ha'porth, for what ever a ha'porth is worth nowadays
