Thats exactly what would happen. They would require your home phone number. I signed up for free army socks a few years ago. Never got my socks but got 2-3 phone calls a week from a recruter. They are sneaky too.
Him: "Hi, this is John. What's up buddy?"
Me: "Um... Do I know you?"
Him: "Suuure. So what are you in to?"
Me: "huh? What are you talking about"
Him: "So do you like mountain biking?"
Me: "WTF?"
Him " I do, we should go some time."
Me: "Who the heck ARE you?"
Him: "Saaaay, You ever think of joining the army? You could do lots of mountain biking at your new post! Think about it!"
Me: "What!! I never said I liked mountain biking. Are you with the Army or something?
Him: "Sure am and let me tell you about it. blah blah blah blah blah blah......."
(Actual conversation I had with an army recruiter a few years ago.)
Now, the military isn't a bad way to go. (I joined the Air Force ) But recruiters have huge quotas they are required to fill each month. Some are more determined than others. I actually had a navy recruiter evesdrop on a conversation I had with the air force recruiter. He heard my home phone number and called me later that day.
So go get your free shirt, but be ready to get lots of junk mail and phone calls.