According to the tax code, "If you and your spouse file separate returns and one of you decides to take itemized deductions (on Schedule A), the other spouse will have a standard deduction of zero."

So, It would it would make more sense for the other spouse to also share in the itemized deductions to lower the taxable income and hence have a lower tax liability.

How do you share the itemized deduction?
Each spouse may be able to claim itemized deductions on a married filing separate return for certain expenses that are either paid separately or jointly with the other spouse.

If the deductible expenses that are paid out of separate funds, such as medical expenses or employee business expenses, they are are deductible by the spouse who pays them.

If the Mortgage payment and Property taxes are paid through a joint account they be shared equally on the Schedule A of the respective Spouses tax return.

Sometimes the deductible expenses that are paid from community funds, the allocation of the deductions may depend on whether or not you live in a community property state. In a community property state, the deduction is shared equally between both of the spouses.

Thus the answer to your second question is yes, both the spouses filing a separate tax return may claim the itemized deductions provided the expenses are shared as discussed above.

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