It's a good thing that you asked how to view private Instagram profiles in 2014, because I have been hacking private Instagram profiles since way back in 2012 (before Instagram was bought by Facebook) and I can tell you that the security has changed a lot throughout this time.
The hacks that I was using to get into private profiles in 2013 just don't work in Instagram any more.
So these are the methods I'm currently using to access private Instagram profiles:
1. You can hack the password of the Instagram account
This will take a few days though... There are quicker ways below.
2. If you are a developer or know how to hack, then read the following article posted on Reddit about using the mobile sessions to expose problems with the API.
This particular bug is patched, but it highlights the kind of security flaw to look out for:
3. Use the oldest trick in the book:
Step 1: Create an Instagram profile based on a friend of the person you want to hack. Copy all their photos and reupload them one by one.
Step 2: Get in touch with the user you want to hack, and tell them you've been hacked and had to create a new profile.
Step 3: Wait for them to accept your new request on Instagram,
Voila! Private Instagram Access