We present what we believe to be the first experimental study of the optical response of collective plasmon resonances in regular arrays of nanoresonators to local environment. Recently observed collective plasmon modes arise due to diffractive coupling of localized plasmons and yield almost 1 order of magnitude improvement in resonance quality. We measure the response of these modes to tiny variations of the refractive index of both gaseous and liquid media. We show that the phase sensitivity of the collective resonances can be more than 2 orders of magnitude better than the best amplitude sensitivity of the same nanodot array as well as 1 order of magnitude better than the phase sensitivity in surface plasmon resonance sensors.