how can access a file for another user in dropbox

The first kind of link you posted, with "/home/" in it, is the private URL of a file in a user's account. It doesn't offer access to the file unless you're logged in to that account.

The second kind of link you posted, with "/s/", is a shared link for a file, which does allow access to the file.

There isn't a way to get a shared link from the private URL unless you have access to the account.

If you do have access to the account, you can get a shared link as shown here:

https://www.dropbox.com/help/167

Or, using the API, using the /sharing/create_shared_link_with_settings endpoint:

https://www.dropbox.com/developers/documentation/http/documentation#sharing-create_shared_link_with_settings