The short answer is no. Writing classes will be more useful when pursuing a career as a Private Investigator because you have to write highly detailed reports for your clients. It helps to start out as a police officer, but it is not a requirement.
I used to be a Private Investigator for a firm that investigated insurance fraud and took a few domestic cases. Becoming a licensed private investigator is a long process, you have to work for a firm or another licensed P.I. for 2-3 years (depending on the state you are getting licensed in) before you are able to apply for your own license and start your own business.
When I applied for my P.I. position I had no law enforcement experience. The qualifications that stood out to the hiring manager were:
Military experience: I had been in the Army Reserves (transportation corps) for 6 years.
Education: I had recently earned an Associate's Degree in Criminal justice from a community college, and was working on my Bachelor's Degree in Criminal Justice from an online school.
Extracurricular Activities: I included my kickboxing classes and martial arts training on my resume. This was of particular interest to the hiring manager because private investigations can be a dangerous job, especially in Baltimore (some of my cases were in rough Baltimore neighborhoods).
To be a successful P.I. you'll need to be creative. You'll have to question a lot of people and you'll have to come up with creative ways of getting them to speak to you. They're probably already suspicious, so you have to catch them off their guard, or put them at ease before you can get the information you need. You also have to be creative with the way you dress. If you're following the same person for several days, you'll have to find ways to quickly change your appearance so they won't recognize you.
You'll also need to be independent (you'll often be working alone, paying two investigators for one case is very expensive). You'll also need to be outgoing, patient, comfortable with working outside in all types of weather, and have a good eye for detail.
Being a Private Investigator can be an exciting career at times, but it also gets very boring sitting in a car for hours at a time waiting for something to happen. It gets really cold, you can't run the engine and warm up the car on a stakeout because the exhaust will draw people's attention and they'll will call the cops on you because they think you're a stalker or something. It can also be dangerous. When people are up to no good they'll get upset and irrational if they discover that someone is onto them.
Ultimately, the boring hours alone in a car made me want to choose another career. I missed really missed working with people and being able to go to the bathroom whenever I needed to.
I hope this insight is helpful to you. Good luck!