The presence of noise allows to exchange ("trade") time regularity for space regularity. For a simple example, consider the map
$$(t,x)\mapsto \int_0^t b(s,x+W_s)\,ds,$$
with Gaussian noise $W_s$. If without the noise the map is $C_t^1C_x^0$, with the noise it becomes $C_t^0 C_x^1$, so the improvement of space regularity is obtained at the cost of loss of time regularity. See these lecture notes (pages 14+15).

In the case of the stochastic heat equation, the noise converts $C_t^{1/2}C_x^0$ regularity into $C_t^{1/4}C_x^{1/2}$ regularity, at a "rate of two powers of space to one power of time". This is derived in section 5.1 of these notes. The heuristics for the two-to-one conversion rate is based on dimensional analysis: The heat equation $u_t=Du_{xx}$ has one dimensionfull coefficient $D$ with dimensions length$^2$/time. This allows the conversion of $p$ powers of space into $q$ powers of time at a ratio $p:q=2:1$.