I'm an Industrial Engineer (which does not require a Professional Engineer license or PE like Mechanical or Civil Engineers need to be), but in my experience it varies. I've had friends that were Civil or Petroleum Engineers that did own their own businesses and ones that work for large and small companies. Given the requirements to become a Professional Engineer (testing, hours working with a licensed PE, etc.) I believe most engineers start out working with a company to meet the licensing requirements then can branch out on their own if they want to. You can also study engineering and then start a business where you don't need to be a PE (something that doesn't require the approval or oversight of a PE).
Hope this helps!
Please check a few open options.
1) Work for Information Technology or other departments in banking, telecom, pharma, retail, etc. domains.
2) Once you gain good experience, you can open your own business/start-up company, provided you have innovative ideas.
3) Work as a professor in colleges
+ there are more open options
Drew Peake, MSME, M.Eng., MBA, PE, FNAFE, DEE, CIH, CSP
I have been in private practice since 1988 and love it. My projects are varying and always interesting. I encourage anyone interested in owning their own engineering business to learn while working under one of the industry exemptions, then once you are sure there is a market for your particular skill set, go for it.
Drew recommends the following next steps:
In disciplines such as civil engineering its more common to find professional engineers (PE's) where they are certified to sign off on on building drawings and such. In this case I would say they can definitely own their own business but they are limited.
Hopefully this answers your question!