pH is defined as minus log of H^{+} activity, so in theory it can exist (and be measured, once experimental procedures are developed) in every solvent. Same about pOH. However, it doesn't follow that knowing one you can always calculate the other. In nonaqueous solutions presence of H^{+} doesn't imply presence of OH^{-}. For example if solution of dry sulfuric acid dissociation still takes place:

H_{2}SO_{4} <-> HSO_{4}^{-} + H^{+}

However, what is really happening is that H^{+} protonates sulfuric acid:

2H_{2}SO_{4} <-> H_{3}SO_{4}^{+} + HSO_{4}^{-}

(and it is not just a theoretical construct, ion product - similar to water ions product - is quite high, [H_{3}SO_{4}^{+}][HSO_{4}^{-}] = 10^{-2.9}) so you have H_{3}SO_{4}^{+} (in a way analogue to H_{3}O^{+}) and conjugate base HSO_{4}^{-}. We can define pH for such solution, but we can't calculate pOH.