Question

How can some manganese oxides be acidic and some basic?

Answer 1

This is not a property only for manganese but also for other elements like Chrome, Bismuth and other "amphoteric" elements. The phenomenon is due to the inductive effect of the several oxygen atoms that are present in the molecule at high value of potential. These oxigen atoms because of inductive effect attract electrons of the molecule leading to release hydrogen atoms (Lovry- Bronsted theory), or show lone pair attractions (Lewis theory).

Answer 2

For the manganese you have MnO and its idroxide `Mn(OH)_2` , `Mn_2O_3` and its idroxide `Mn(OH)_3` that have basic features. `MnO_3 and Mn_2O_3` that are acid oxides and give the acids `H_2MnO_4 and HMnO_4` that are quite strong acids.
`MnO_2` is an oxide with weak acid property and its solubility is very little.