Photobucket is what I use. Here are some instructions. Hopefully they make sense!
1. Go to your Photobucket library and click on the image you want to share.
2. On the right hand side there are a list of different options under the title "Share Links"
3. Choose "direct" and click in the box. The URL will be automatically copied.
4. Go to the post you want to put an image in on Psychcafe and paste the link into the post.
5. Put the following tags in square brackets in front of and at the end of your link. [ img ] and [/ img]
Note: I have put spaces on either side of the img code so you can see it. You don't need any spaces. So, if we replace square brackets with curly ones your URL would look like this:
Hope that helps.
On another note, regarding privacy, it's good to keep in mind that anyone can work out the source code for your image and therefore find their way back to your Photobucket account and browse through any pictures you have set to public. For that reason I keep a separate Photobucket account for forums and only put pictures in that I would be happy for the general public to see.