First, you really should be asking this on https://superuser.com. There are a lot more privoxy questions there.
Yes this will work and yes you can have it on the same machine.
There are three moving parts here and this post is going to be way too long if I don't freeze one of them so let's just say you're running a straight-out-of-the box vanilla privoxy, bound to 127.0.0.1 ipv4 only, port 8118.
The important openvpn options (in addition to whatever hidemyass gives you)
route-nopull and one of
redirect-gateway def1 or
route-nopull stops your client from accepting routes from hidemyass (they'd probably just hand you a default route, there's not much else they could do).
redirect-gateway def1 gives you a route to 0.0.0.0/1 with hidemyass as your gateway: it is slightly more specific than your existing default route on your LAN (it has a smaller mask)
the second option,
redirect-gateway block-local which I would imagine you want, will block traffic to the LAN (except for VPN) while you are connected to the VPN. It does this by adding a route to 0.0.0.0/1 and a route to your LAN subnet with hidemyass as your gateway. You would not use this if you wanted to provide proxy services for other computers on your LAN, and you would configure privoxy to listen to your LAN ip address AND lo interface.
So all browsers (LAN or just you, whichever setup you prefer) that have been configured to use your local IP:8118 as a proxy server. You aren't actually doing any routing or packet forwarding here (proxying is not routing).
The third moving part, routing other stuff, I can't even address because you didn't tell us what OS you're running beyond Chrome and IE as clues. So what OS are you running?