AirNoServices:
I'll tell you what happens in one of the very few pieces of Class E in the UK, where I fly VFR regularly and IFR occasionally. The airspace in question is part of the Scottish TMA and extends from 1500 or 2500 ft up to 6000ft, all around the boundary of the Edinburgh Control Zone (Class D).
When we're outbound from Edinburgh VFR we get a clearance not above 2000ft. IFR inbounds are routinely cleared to 3000ft crossing over the top of us and they are always given traffic info on us. We are normally transponder-equipped and squawking 7000 but even if no transponder the clearance and procedure is the same.
When we clear the zone we are in Class E airspace and in theory we could manoeuvre and climb unrestricted and without saying anything to Edinburgh Approach, but usually if Approach has IFR inbounds they'll say "report if you wish to climb above 2000ft". If you say you do want to climb, some controllers will ask you to maintain 2000 for the moment due to IFR inbound at 3000. It's a grey area because one might argue that by acknowledging that request one is agreeing to comply, even though we are only receiving a Flight Information Service. We could legally climb VFR to whatever altitude we wanted without telling Approach, but common sense usually prevails. Approach do however get the occasional unknown aircraft flying into the airspace VFR - again, quite legally - and causing mayhem with inbound IFR traffic.
My limited experience in the US is that in equivalent airspace, controllers will not try to maintain 1000ft/3nm separation between VFRs and IFRs and will rely on the crews seeing and avoiding each other. In the UK they wouldn't do that.
Having said all that, this Class E airspace is about to be abolished because of an airprox a couple of years ago between a 737 and a microlight both of which were operating quite legally in that Class E airspace.