I don't think it's a good idea because for one, it's not facebook. It's your own community. If you don't have the same friends existing on your network as on Facebook, then what does it matter to have those posts, or likes, pictures or anything else that resides on your Facebook page? I think it's better to start fresh considering you have to have your friends on your new site. The outcome will be broken up conversations with people that don't exists, likes on people that don't exists and a lot of one sided conversation. Everything you do on Facebook pertained to who you know or knew on Facebook and what was occurring on Facebook.
This is a fairly hot topic and we have been trying to approach it in different ways. Facebook is just one network... there's also Twitter, LinkedIn, regional networks, etc. Ideally, you'd want some way to allow people to download their data from those network and import it to your site, with some level of control in selecting what to upload and what not to upload. It has to be be fully controlled by user. So, there's has to be some mediator that allows to easily download the data, normalise it into readable API/format and allow us to import with approval. We have been in contact with digi.me - a service that does just that, and so far we're at the stage where we need to "match" as much FB/Twiiter/LI/etc functionality as possible to be able to accept that data. Configuration of ONLINE.ME is part of that journey, but I would expect at least 12-18 month until it becomes a reality.
I understand what you mean, but it's just data we want. e.g. someone has 10 years of Photos events and stuff on FB or twitter or so... we can aggregate all at one place and then maybe user will never go to those sites, if they like what it here. It could be visible to only the owner for the moment and then they can make some of it available to public as Andrew mentioned below.