Either. As noted you will need private proxies to post with and make sure you have them enabled for posting or your ISP would wind up with complaints. As for your internet connection you would need enough bandwidth. Like I have a 105Mbit package at home and I can run a lot of threads on that.
DNS plays a huge part. My 105 pack is from comcast who may have one of, if not the worlds worst DNS. I could just about write down my GET and POST requests and send them snail mail and get printouts back as a reply faster. hehe
So you can use a public DNS like Google or Level 3 or ideally both. This works for me. I run a couple boxes in my basement from time to time and use google/level 3 for DNS and then I run my standard web surfing just thru comcast dns.
Bear in mind if you get into high connections not only will DNS be a concern, your router will play a part. The $60 linksys router I used to have back a few years ago would not handle 300+ connections, even a $60 gigabit wireless N router right now won't likely handle it well. I picked up a $150 Asus router that worked wonders and they have nicer more expensive ones out now. Plus the wireless signal was exponentially better as well, an added bonus.
So lots of things to keep in mind. I have many dedicated servers, but they all cost so sometimes rather then buy another I use old machines in the basement that I already have on an internet connection I already have with a router I already have which costs nothing more then my time to make sure the whole thing doesn't fall apart from time to time.
All in all I favor servers as they are no maintnance on my end and I can remote into them on my ipad and pc easily to control them. I even can RDP servers while on flights and pretty much anywhere I have a connection, love my ipad for this for mobile and laptop is great when at a desk for not in a plane/car etc...
Anyway, its a cost no matter how you slice it. Running it in house will take more of your time and less of your money (at least long term, generally speaking), running it as a vps will be less time but more money long term. So it just depend on what you have more of to spend, time or money. I mostly have more money then time so I opt for servers, which is why at the moment the basement servers are off collecting dust.
So what do you have more of to spend? Worst case start slow and at home and you have no cost except proxies and then just upgrade to a vps as you need it.