Like many have said, it largely depends on a number of circumstances. Posting an image takes up bandwidth. If you have a host that gives unlimited up/down bandwidth and provides a really big pipe, this may not be an issue. If you’re limited this could be a problem, more costly than anything else.
The costs of building a site that handles upload of images, downloads, posts on them, possibly authenticated sharing (so that your users may want to control who sees the images) could be a bit expensive, but if you start small and only have a few users, and some how make some money (or if your rich and don’t care about that), then a hosted solution probably would be fine.
If you are planning on growing, maybe working with other services like flickr, facebook and the likes, you’ll probably need some custom development. Most important on this is figuring out what you want to go with… php, java, .net. I myself am a java guy. Most people seem to find hosted solutions with php.
It also depends on what sort of funding you have to pay for all this. You of course have to manage more than just the hosting cost, getting help if need be, extra features, etc.
Sadly, many people want to start a site and get suckered in to the bid sites, where they hire a guy in india for $7 to build them a site for a month. What almost always results is nothing close to what was wanted, and you’ve wasted a lot of time… and time is money. My costs are quite expensive, but then, I know how to set up large scale sites from the front end UI/web pages, to the back end database, transactions and so forth. It’s not cheap, but I have a lot of experience with it, so you get what you pay for.
So really it depends on a lot of what you need. Do people log in, make connections to other friends… or do they just post a pic and allow the world to see it? Is there any login required at all, or is it for anyone to post anything for anyone to see? The list goes on.
Good luck.