You can file together. You do not need to file separately. If you file separately, yes you would both claim half of the mortgage interest. If one itemizes, you both have to itemize.
If one uses the standard deduction, then both use the standard deduction.
It is almost always better to file jointly. There are some special circumstances where it is better to file separately.
Here is a great article explaining the advantages of filing jointly and the consequences of filing separately. It also gives you some instances when it might be better to file separately.
Filing Jointly vs Separately
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