My Dropbox shared folder is not syncing to the second computer

If that user has set the shared file/folder to sync to their computer then yes, it will stay updated when their dropbox syncs.

Keep in mind, though, that there are many ways that the file/folder could be out of sync:

• Other person hasn't turned on their computer
• Other person hasn't turned on syncing for that file/folder
• Other person has Dropbox setup to not auto-sync
• Other person has that file open and Dropbox can't update the file because it's in use.

I had a similar issue, but in my case it was purely a misunderstaning of the Dropbox UI.

When you click the link in the email you get when someone shares a folder with you, there is a "Download" button up at the right on the website you come to (see this screenshot). I thought that button meant that I would download the files once to my local drive – while that is true, it turns out it was also the way to add the folder to your Dropbox.

In case you too missed the function of that button, just click it and select "Add to my Dropbox" (again, see the screenshot).