"The man is believed to have escaped in a stolen car." That is more correct. It means that someone, or some people, believe that he escaped in a stolen car. It's a weird sentence construction, but very common in news articles.

The first one is correct :)

It is believed that the man has escaped in a stolen car. Or, more simply: The man escaped in a stolen car. The first version is verbose, wordy, it has too many words but is correct. "is believed" is strange and a construct that exists in a lot of other languages and is grammatically correct but strange in English.

The first one is correct because the verb tenses agree. In the second one you are using two different word tenses.