I think the variance of the instantaneous shifts in the spread is meant:
$V \left[ dX \right]=V \left[ dS_1-dS_2 \right]$
And the individual variances (in the conditional and local sense) are:
$V \left[ dS_1 \right]= \sigma_1^2 S_1^2dt$
$V \left[ dS_2 \right]= \sigma_2^2 S_2^2dt$
And the covariance term is, assuming the two Brownians are correlated:...