"classically trained" can mean many things...what sort of repertoire do you play?
jazz means many things as well...what sort of jazz interests you?
different modes require different approaches.
if you want to play straight jazz--you know, covers of bop tunes, etc.---then you'll need to learn scales and how to use them as guides for making lines.
george russell's book is excellent as a place to start (and will put you outside the usual run of things at the same time):
http://www.lydianchromaticconcept.com/
if your training is more geared around contemporary classical music, you may find it easier to use a different, less traditional approach. if you can get ahold of anthony braxton recordings and/or his writings (or his compositions) check them out.
this will give an idea:
http://www.wesleyan.edu/music/braxton/
without any further information, i'd suggest 3 basic things:
1. try working with chord structures that do no use thirds. close intervals or spread intervals allow for more space. seconds or tenths. try to think about building lines rather than building riffs. classical training gives you ways to think about building lines--the basic tools of melodic variation--a phrase, its inversion, its retrograde, its retrograde inversion. use them. dont get stuck on doing riffs--they limiting and limited--besides, if you think in terms of architecture, you can still do them.
this because you should get away from the idea of right or wrong notes, i think. there are no right or wrong notes--what you play is held together by its structure.
the structures you can use are a function of the context that pleases you to work in.
2. straight jazz players tend to play one handed--block chords in the left and riffs in the right. this is not necessary. i'd suggest listening to stride players like james p johnson to hear how he uses the tenths as interior melodic elements. listen to thelonious monk for everything, though monk was so good that it can be dispiriting--duke ellington in small group contexts as well (stuff like "money jungle"); andrew hill for a hybrid type of playing that combines block chords and counterpoint; cecil taylor.
3. play with other people.
o yeah--this is important:
people like to make doctrinaire claims based on the type of music they like.
listen to them, but dont take them seriously.
if someone says that you *have to* play scales to play jazz, then think about it--what form are they working out of? outside that form, the imperative means nothing.
most important: the only way to learn to improvise is to improvise. it doesnt matter if at first it sounds random. it'll change as your thinking changes. your thinking will change through doing. play alot. have fun. dont worry so much.