"I used to be asked all the time, 'Did you ever run away?' No, I never ran away … I respect that they had a job to do. I wasn't going to make that job harder for them, and I wasn't going to stress out my parents," says the former first daughter and In Fact podcast host

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Chelsea Clinton is reminiscing about growing up in the White House.

The former first daughter opened up about spending her teen years under the watchful eye of the Secret Service and creating lasting traditions with her family, despite their busy schedules.

"Thankfully nothing that dramatic ever happened to me," she told Janine Rubenstein during an appearance on Wednesday's episode of the PEOPLE Every Day podcast.

"I once was driving with my best friend from high school and we went through a light and like the Secret Service got caught behind the other light and we pulled over immediately of course, and waited for them," said Clinton, now 41 and the host of her own In Fact health podcast. "But I was like, 'Oh my God, like I'm outside of their immediate line of sight for like two minutes.' And I think I apologized for two weeks. I felt so horrible."

She continued, "I used to be asked all the time, 'Did you ever run away?' No, I never ran away ... I respect that they had a job to do. I wasn't going to make that job harder for them, and I wasn't going to stress out my parents [Bill and Hillary Clinton] like that. Just never even occurred to me."

Listen below to the episode of our daily podcast PEOPLE Every Day for more on Chelsea Clinton's White House childhood and her new public health podcast.

Chelsea Clinton
Chelsea Clinton
| Credit: Bennett Raglin/Getty Images

Chelsea moved into the White House just before her 13th birthday.

"I have so many wonderful memories ... most of them center around the extraordinary people who are part of the White House, who were there regardless of who the president is, regardless of the current administration," she said. "So the White House curators and the White House butlers and the chefs and the florists and the arborists and the historians, are just an amazing community of people. And I learned so much about the history of certainly the White House and also the history of our country."

She said on PEOPLE Every Day that even though she was a teenager, her parents were still "very protective" of her and made sure to carve out time every day to spend together.

"We treated family dinner as very sacred time," the mom of three explained. "Like my parents often would have to go back to work after dinner, but even when I was a teenager [and] so didn't need as much, kind of physical care ... that still was just a really kind of sacred family time."

That tradition is one that Chelsea and her husband, Marc Mezvinsky, have remained "very much committed to" in their lives with their kids — Aidan, 4,  Jasper, 19 months, and Charlotte, 6.

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"We have a lot of creating our own stories as a family, and all of that we did too when I was growing up," she said.

Last month, Chelsea launched In Fact, focusing on public health. She has already tackled several heavyweight topics, including HIV and AIDS, climate change and substance abuse.

"I just have never seen so many people paying attention to public health as what we're all living through right now, understandably, because of COVID-19. And yet, I don't think a lot of people know that public health isn't just infectious diseases," Chelsea said. "Public health is affected by climate change, by gun violence, by substance use disorders and addiction. I think so many people tend to think gun violence or substance use disorders are kind of individual tragedies, challenges. And yet, they're really community tragedies and challenges.

She told PEOPLE that her interest in health policy began while she was a child in the early 1990s, not long before her father was elected president.

Hillary and Chelsea Clinton
Credit: Brooks Kraft/Getty

"I really remember being a kid and watching Magic Johnson talk about being HIV-positive," she said. "And I remember exactly where I was when I watched the clip of his press conference on the nightly news as a kid in Arkansas. And I just got really, I think, intellectually and also emotionally interested in the AIDS crisis and in the real deep inequities in the AIDS crisis in this country, and then globally."

With her own podcast — that has included guests such as Dax Shepard and Amber Tamblyn as well as experts like Peter Staley — Chelsea said she hopes people can "listen to conversations that might be interesting to them and yet they'll come away understanding how they relate to public health. Hopefully a little more informed and hopefully a little, or maybe a lot more, inspired to engage on these issues."

The past year has been a busy one for her. She released her children's book She Persisted in Sports in September.

Last March, Chelsea also published a picture book along with her mother called Grandma's Garden. The book was inspired by the former first lady's late own mother, Dorothy Rodham, who loved to read and garden with them both.

In Fact releases episodes every Tuesday.