I am getting ride of some old magazines, Popular Science/Mechanics, Maxim. All of these magazines of my name and Address on them. The address is no longer my address but my name is the same. Can anything bad happen or should I sharpie out 2 years worth of all the magazines?
Some people worry more about this than others. I’m on the side that doesn’t worry at all. Your name is no secret, it’s not your address anymore (and your address was probably not secret either). Anything “they” can find out about you from these old magazines, can be found out in other ways.
Worst thing I’ve ever been able to imagine is that some malicious person could call up the magazine’s subscription department and use the account number/name/delivery address info to convince them they were you and stop/restart/change your subscription. Or get your new address.
But identity theft prevention folks say not to toss anything out that has your name and address on it, so I’d take a pair of scissors to them before tossing them in the recycling. Not a Sharpie, though: my experience is that the ink they print those addresses on will read right through a Sharpie.
Which usually makes me try to figure out what the name and address was, instead of completely ignoring the address block.
Seriously, I’ve never understood why people worry about that. I have a coworker whose husband burns every single piece of mail with their address on it “in case people steal it”. If they stole it from your trash can, don’t you think they’d know your address? Of course, she’s not the sharpest knife in the crayon box as it is.
I mean, what could I do with your name and address? I have delivered to me free every year whether I want it or not a big yellow book FULL of names and addresses.
Worst thing that can happen is that your magazines end up in a dumpster, and a drug smuggler decides it’s better to get rid of the evidence rather than to get caught. Later when the police finds a few pounds of heroin hidden in a bag along with your name in it, you’ll have a drug enforcement SWAT team break down your door, shoot your dogs, point machine guns into your children’s faces, and turn your entire home upside down faster than you can say “Why me? Can’t we just sit down and talk about it, I’m sure this is all a misunderstanding.”
Seriously, and I’ve never understood why people don’t worry about leaving needless traces. Even if the chance is remote, why risk it if you can avoid it? Just because you’re not paranoid doesn’t mean they’re not out to get you… 
Or how about a multi-million dollar lawsuit? I mean, we’re living in a world where the content industry argued that throwing a promotional CD in the trash is an unauthorized distribution and a copyright violation. UMG Says Throwing Away Promo CDs is Illegal | Electronic Frontier Foundation
Well, if you’re ok with people knowing you read Maxim I wouldn’t worry about it.
I work for a mega-corp and I have the highest security clearance within the company to see critical data for millions of people including people I grew up with, people I know now, my own coworkers, and maybe even someone in this thread. You might think that would make someone paranoid but it is just the opposite. Some of my own coworkers have the same clearance and I have no doubt they have done the same to me.
Names and addresses are trivial information. We have to undergo extensive HIPAA training for the really private stuff but names and addresses are not it. Names and addresses float around among thousands of companies as mailing lists are bought and sold. You have to get down to individual Social Security Numbers before it becomes a concern. Just shred or rip up your bank and credit card statements and you will be fine.
Really ripping up financial statements is needed. A good shredder is better.
As a general rule of thumb, I shred anything that involves a company that I have a financial arrangement with. So banks, CCs, utilities, etc.
I don’t worry about junk mail unless it’s of the CC offer variety.
As people routinely point it, it is quite easy to social engineer people at companies into changing address or other info in order to take over an account. A SSN is hardly necessary to do harm.
One experience of mine. At the last place we lived, we cancelled our newspaper subscription before we moved. The new owners then re-opened the account in our name. After a year or two we got a bill at our new place. Had to call the idiot newspaper up to point out: We cancelled, we sold the house, maybe you should be billing the people that have lived at the place you’ve been delivering the newspaper.
Note that this comes close to the issue of the OP. Someone gets access to an old magazine label. Calls up the magazine, pretends to be you, renews but gives a new address. Then later the magazine comes looking for you to pay up. It happens. So rip off the address label. Forget using a marker.
Companies are stupid, employees are poorly trained. E.g., I recently heard there’s a company training people in security that thinks it requires at least a SSN to steal an identity.