Die-hard fans may love the freewheeling ways of Tesla CEO Elon Musk, but when it comes to running a public company, his behavior just isn't appropriate, corporate governance expert Betsy Atkins told CNBC on Wednesday.

"People love Elon Musk because he's an independent thinker, but as a public company CEO, he is a train wreck," Atkins said on "Squawk on the Street."

Musk rattled markets on Tuesday after he announced in a string of midday tweets that he was considering taking the electric car maker private at $420 per share, and that he already had funding secured. Tesla's stock jumped as high as $371.15, after the first tweet, but was later halted. Musk released a blog post later Tuesday, discussing the tweet in depth. Tesla resumed trading, closing Tuesday at $379.57. Several of Tesla's board members released an official statement on Wednesday confirming the news and saying the board met last week to discuss and evaluate the decision.

Atkins, who has served on the boards of several public companies including Wynn Resorts and Volvo, said Musk already behaves like he's the CEO of a private company — which is to say he is much more inclined to take risks.

Private company chief executives are allowed that type of freedom, but public company CEOs must think first about shareholders and protocol, according to Forbes.

Robert Nardelli, former chairman and chief executive officer of Chrysler, emphasized that private companies are still beholden to the same standards of performance as public ones.

"Believe me, in private equity, there's still the same expectation that there's going to be profitability, that the business has solid performance, predictable and quality of earnings," he said Wednesday on CNBC's "Squawk Alley."

There are some definite benefits from private equity, like the ability to quickly make decisions and sell nonearnings assets for cash "which certainly Tesla is in need of," Nardelli said.

"There were clearly advantages I saw on the private side with Chrysler, not that there was any less pressure on performance, getting production out, quality of product ... all the things that Elon will continue to have to deal with," he said.

Vivek Wadhwa, Carnegie Mellon professor and co-author of the book "The Driver in the Driverless Car," defended Musk, and said that taking Tesla private is "the best thing" for the CEO right now.

"He gets control of the company and he can innovate. It frees him up to do what he's best at which is innovating and changing the world," Wadhwa said on "Squawk on the Street."

"As far as his tweets go, we are in the era of Donald Trump. So what if he's tweeting? Everyone does it and that's how we communicate these days," he added.

Atkins, however, expressed concern that the tweets went beyond making strong pronouncements to violating the Securities and Exchange Commission's regulations on fair disclosure, which prohibits public companies from disclosing nonpublic, material information to certain individuals or entities unless the information is also disclosed to the public.

The board, which also "behaves a lot more like a private company board" is enabling Musk's behavior, Atkins said.

"This board's oversight is very odd. They knew they had a cowboy CEO. He's baited the analysts, he's tweeted before, they should have reeled him in," said Atkins, who is also CEO of Baja Corp. "They should know they have a rogue here and they know very well how you have to communicate with the market."

Musk's recent behavior and the board's lack of oversight may open up the company to lawsuits and investigations, Atkins warned.

"They will likely get investigated about how they released information with a tweet," Atkins said.

Tesla stock was last down 1.6 percent on Wednesday at about $373.50 per share.

Tesla did not immediately respond to CNBC's request for comment.