  Forward self-similar and discretely self-similar weak solutions of the
Navier-Stokes equations are known to exist globally in time for large
self-similar and discretely self-similar initial data and are known to be
regular outside of a space-time paraboloid. In this paper, we establish spatial
decay rates for such solutions which hold in the region of regularity provided
the initial data has locally sub-critical regularity away from the origin. In
particular, we (1) lower the H\"older regularity of the data required to obtain
an optimal decay rate for the nonlinear part of the flow compared to the
existing literature, (2) establish new decay rates without logarithmic
corrections for some smooth data, (3) provide new decay rates for solutions
with rough data, and, as an application of our decay rates, (4) provide new
upper bounds on how rapidly potentially {non-unique}, scaling invariant local
energy solutions can separate away from the origin.
