- The chef is Italian, do you think?
This is grammatical, but perhaps on the rarefied side of formal. Perversely, the tag-like (true tag questions don't include lexical verbs like 'think' or 'say' ['would you say?']) version of 'do you think?' (meaning 'in your opinion') is also used colloquially:
“Whose is it, do you think?" I say finally.
"No telling," says Finnick. "Why don't we let Peeta claim it, since he
[Suzanne Collins, Catching Fire]
Here, "Do you think the chef is Italian?" is idiomatic.
But "The chef has to be Italian, wouldn't you say?" is a perfectly idiomatic example of a similar-looking type of question structure. Correspondingly, "The chef has to be Italian, don't you think?" uses the correctly structured tag-question (negative after a positive assertion), but this is less of a query than "Do you think the chef is Italian?" and more of a coercing "You have to agree with me".
- The chef is Italian, you think?
This too is grammatical, but would only be used to confirm an implication or assertion from a previous statement (such as "This cooking is authentic Italian, not the imitation we sometimes get in 'Italian' restautants."