Is it a service page or a landing page? A landing page is meant to convert a lead. Typically you drive traffic to that kind of page. So if the target audience is the same for both landing pages, I doubt it would deter people.
There are huge agencies that use duplicate content for business across an industry. If the page is built well, the content reads, and there isn’t much local competition, the page will still rank. I hope this helps a bit!