List<String> sObjList = new List<String>{'Account','Contact'};

Schema.DescribeSObjectResult[] descResult = Schema.describeSObjects(sObjList);
Map<String, List<String>> objToRequiredFieldsMap = new Map<String, List<String>>();
for(Schema.DescribeSObjectResult dsr : descResult){
    List<String> requiredFields = new List<String>();
    //get required fields
    Map<String, Schema.SObjectField> fieldMap = dsr.fields.getMap();
    for(String str : fieldMap.keySet()){
        Schema.DescribeFieldResult f = fieldMap.get(str).getDescribe();
        if(!f.isNillable()){
            requiredFields.add(f.getName());
        }
    }
    objToRequiredFieldsMap.put(dsr.getName(), requiredFields);
}
System.debug(objToRequiredFieldsMap);