I have seen question Embedded surface in simply connected manifold is two sided? and I was wondering if the same result holds if the hypersurface is only immersed and not embedded i.e.
let $\Sigma$ be an immersed hypersurface of $M$ simply connected. Is it true that $\Sigma$ is two-sided?
Consider Boy's surface, which is an immersion of the projective plane into 3-space. If it were two-sided, then the normal bundle would be trivial, so $w_1$, the first Stieffel Whitney class would be $0$ (by the whitney sum theorem, or whatever it's called), so the projective plane would be orientable.
So the answer to your question is "no", because there's an immersed surface in $\Bbb R^3$ (or $S^3$) which is not two-sided. (BTW, I'm using here the notion of 2-sided-ness which is "normal bundle is trivial"; it's not obvious to me how to extend other definitions to the immersed-manifold situation).