This paper shows introduces the use sensing dictionaries for p-thresholding, an algorithm to compute simultaneous sparse approximations of multichannel signals over redundant dictionaries. We do both a worst case and average case recovery analyses of this algorithm and show that the latter results in much weaker conditions on the dictionary, sensing dictionary pair. We then do numerical simulations to confirm our theoretical findings, showing that p-thresholding is an interesting low complexity alternative to simultaneous greedy or convex relaxation algorithms for processing sparse multichannel signals with balanced coefficients, and finally point a connection to compressed sensing exploiting the additional freedom in designing the sensing dictionary.