Horizontal points of smooth submanifolds in stratified groups play the role of singular points with respect to the Carnot-Carathéodory distance. When we consider hypersurfaces, they coincide with the well known characteristic points. In two-step groups, we obtain pointwise estimates for the Riemannian surface measure at all horizontal points of submanifolds with tangent spaces of Lipschitz regularity. As an application, for the same class of submanifolds, we establish an integral formula to compute their spherical Hausdorff measure. Our technique also shows that more regular submanifolds admit everywhere an intrinsic blow-up and the limit set is an intrinsically homogeneous algebraic variety.