If my husband and I are filling separate taxes do both of us need to claim our dependents ?

[xmasbaby0]

No.   A dependent's Social Security number can only be used on one tax return.   You cannot double dip and each claim the same child on separate returns.  So....decide who is claiming them or divvy up the kids and each claim one of them, etc.

Why do you want to file separate returns?  That is usually the worst way to file especially if you have children.

If you were legally married at the end of 2022 your filing choices are married filing jointly or married filing separately.

Married Filing Jointly is usually better, even if one spouse had little or no income. When you file a joint return, you and your spouse will get the married filing jointly standard deduction of $25,900 (+$1400 for each spouse 65 or older)  You are eligible for more credits including education credits, earned income credit, child and dependent care credit, and a larger income limit to receive the child tax credit. 

If you choose to file married filing separately, both spouses have to file the same way—either you both itemize or you both use standard deduction. Your tax rate will be higher than on a joint return. Some of the special rules for filing separately include: you cannot get earned income credit, education credits, adoption credits, or deductions for student loan interest. A higher percent of your Social Security benefits may be taxable. Your limit for SALT (state and local taxes and sales tax) will be only $5000 per spouse. In many cases you will not be able to take the child and dependent care credit. The amount you can contribute to a retirement account will be affected. If you live in a community property state, you will be required to provide additional information regarding your spouse’s income. ( Community property states:  AZ, CA, ID, LA, NV, NM, TX, WA, WI)

 If  you are using online TurboTax to prepare your returns, you will need to prepare two separate returns and pay twice.

https://ttlc.intuit.com/questions/1894449-married-filing-jointly-vs-married-filing-separately

https://ttlc.intuit.com/questions/1901162-married-filing-separately-in-community-property-states

https://ttlc.intuit.com/questions/1894449-is-it-better-for-a-married-couple-to-file-jointly-or-separ...

[Mike9241]

if both parents can claim the child, then the IRS says the parent who can claim them is

the parent who the child lived with the most nights during the year

if that's a tie then the parent with the highest adjusted gross income.

as stated MFS is usually worse than filing joint. 

MFS you will lose certain credits and certain deductions may be restricted. also, both must use either the standard deduction or itemize.