You might run into conflicting answers because there are actually several completely different definitions of the stress tensor.
Definition #1
In continuum mechanics, the stress tensor of a solid or fluid is defined in terms of the force that solid/fluid elements exert on their neighbors along an infinitesimal surface element $dS_i$, by
$$dF_i = \sigma_{ij} \, dS_j.$$
This has an overall sign ambiguity, because you can orient the surface element in either direction. An equivalent way to say this is that you can define $dF$ to be the force exerted by the small element you're considering, or the force applied on that element. However, I think the most common choice is to let positive pressure correspond to negative diagonal elements $\sigma_{ii}$.
Definition #2
On the other hand, the first time a physics student typically sees the stress tensor is in an intermediate electromagnetism. Here, the first definition doesn't make any sense, because the electromagnetic field doesn't exert any forces on itself; it only exerts forces on matter. So instead, the Maxwell stress tensor is defined in terms of the force per volume $f_i$ that the electromagnetic field exerts on matter,
$$\partial_i \sigma_{ij} = f_j.$$
In my experience, the sign is universally chosen to be this one, and this time there's no ambiguity with surface elements. Note that this definition agrees with the first, in the sense that when an electromagnetic field exerts a positive pressure on matter, the diagonal elements are negative.
Definition #3
An even more fundamental definition of the stress tensor is that it's the spatial part of the stress-energy tensor $T^{\mu}_{\ \ \ \nu}$, which is defined via Noether's theorem with unambiguous sign. This definition completely abandons the idea of stress meaning anything exerting a force on anything; instead, $T^{\mu}_{\ \ \ \nu}$ just denotes the rate of flow of the conserved quantity $p^\mu$ along the $\nu$ direction, which can occur whether or not anything experiences forces. (For instance, the previous two definitions wouldn't make sense for a gas of completely noninteracting particles, but this one does.)
Unfortunately, in this case there's another sign ambiguity, which is which elements to identify as the "spatial" parts. In Euclidean space, we don't distinguish between up and down indices, so $\sigma_{ij}$ could equally well stand for $T_{ij}$, $T^i_{\ \ j}$, or even $T^{ij}$. There's also a further sign ambiguity due to the signature of the metric.
While conventions can differ, I think that in this context, I've mostly seen the spatial components defined so that positive pressure corresponds to positive diagonal elements, which is nice-looking but unfortunately in conflict with the earlier two definitions. For example, Wikipedia does just this for the electromagnetic stress-energy tensor, resulting in $T^{ij} = - \sigma_{ij}$.
