Statement I: BeSO$_4$ and MgSO$_4$ are readily soluble in water due to the high hydration enthalpy of the smaller Be$^{2+}$ and Mg$^{2+}$ ions, which overcomes their lattice enthalpy. This statement is correct. Statement II: All nitrates of alkaline earth metals decompose on heating to give their corresponding oxides, NO$_2$, and O$_2$. So, this statement is incorrect. It's not exclusive to Be(NO$_3$)$_2$.