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The code has a few bugs.

• In talker(), you've got an infinite while() but you don't really do anything there (pub.publish() isn't publishing any meaningful content).
• Why do you have both while not rospy.is_shutdown() and rospy.spin()? Also, rospy.spin() is unreachable in your code.

Based on message_filters Python example, I re-wrote your code (code is untested):

def callback(data1, data2):
    rospy.loginfo(rospy.get_caller_id())
    ## do stuff with data1 and data2 ##
    pub.publish("some_string")


if __name__ == '__main__':
    rospy.init_node('SubPub', anonymous=True)
    pub = rospy.Publisher('Publisher', String, queue_size=10)
    sub_1 = message_filters.Subscriber("chatter", String)
    sub_2 = message_filters.Subscriber("sub_chatter", String)
    ts = message_filters.ApproximateTimeSynchronizer([sub_1, sub_2], 10, 0.1, allow_headerless=True)
    ts.registerCallback(callback)
    rospy.spin()