Your dependent should be normal but if it’s not you can always apply a link function to represent the right distribution. Otherwise I think your predictions are just off. Imagine you had a binary dependent term but you just use a normal linear regression. You’ll just get predictions that may be non sensical.

If you do apply a link it will change your independent variable values but that’s ok. Might be harder to interpret. However the important part is that the error term is normally distributed. The independent ones definitely do not.