A separation is an informal state of affairs. Your husband has little recourse to force you to pay half of the mortgage unless he goes to court for a divorce. If you do go to court, your state laws and personal financial situations will determine the court's decisions on your mortgage. Meanwhile, your bank will want the mortgage payment and can go after anyone who's name is on the mortgage to get it.
Determine who's name is on the mortgage. The bank will consider both of you responsible for the mortgage payments if both of your names are on the mortgage. In this case, if your husband does not make the mortgage payment, the bank will come to you looking for payment. Refusing to pay will damage both of your credit scores and can result in more aggressive collection efforts and eventual foreclosure.
Check whether your state considers the equity in your home to be communal marital property. If you will eventual seek a divorce, it may be in your best interest to continue paying half the mortgage to ensure your share in the house. In communal property states, you will receive half the value of the equity in the house, either from sale of the home or a buy-out by your husband, at the end of the divorce process. However, your share of this equity could be reduced if your husband shows that his personal funds, rather than marital funds, were used to make mortgage payments.
Official determination of whether you must pay half the mortgage will not come unless you file for divorce. In this case, the divorce court will be the ultimate arbiter. The court will consider the equity, who's name is on the deed and mortgage, and your individual financial situations when determining who will be responsible for the mortgage. The court is more likely to force you to continue making mortgage payments if you were the primary, or only, earner during the marriage.
Prior to an official divorce, who makes the mortgage payment is largely a personal matter, though the bank will hold you both equally responsible. Get any financial agreements between you and your husband in writing. Consult a lawyer for the particular laws in your state and before signing anything proposed by your husband. Remember that you must look out for your own interests.