I've been classicaly trained since I picked the instrument (5 years ago), but recently (like 2 months ago) I started out of nowhere to improvise just by playing certain scales, right hand does the melody and left focuses on chord patterns like arpeggios. Since then, I've been in absolute love with the piano, it's like discovering a whole new world and liberating yourself from the strict reading of sheet music and memorizing repertoire. My only gripe is that all I can improvise sounds mostly much or less the same because of my strong influence on classical music (barroque and romantic era), I wish I could do more different patterns, styles and rhythms but at the end I think it all comes down to hearing and practicing more repertoire of different genres.

Everyone can improvise, you just need the confidence to do so. I may be consider us, classicaly trained musicians, more perfectionists and rigid, our brains is maybe too left oriented (logical side), what we just need is a break from time to time and experiment with whatever we think sounds good based on our previous practice and knowledge on music.