The influence of exchange interaction between exciton and magnetic impurity spins on magnetic moment and its fluctuations in semimagnetic quantum dots has been investigated with the use of individual dot magnetophotoluminescence technique. In quantum dots with strong exchange interaction, both the polarization of magnetic ions and magnetic fluctuations in a dot are highly influenced by the formation of exciton magnetic polaron. In contrast, in the quantum dots with weak interaction, the localized exciton acts as a perfect noninvasive probe: it allows one to analyze the quantum-dot magnetic moment and its fluctuations with negligibly small influence on the dot magnetization. Numerical values of exchange magnetic field and magnetic polaron energy are obtained and analyzed for both types of dots.