The Watch is designed to handle all the notifications on the phone, without the phone having to give any alerts. The only exception to this would be for phone calls, when the iPhone either rings or vibrates, as well as the Watch. The only way for you to get your phone to vibrate on some notifications is to customize the notification settings on the Watch to only show the notifications you want on the Watch--e.g., Messages, Mail, and Calendar--while turning off all other notifications. If you do this, a notification from Twitter will vibrate the phone, but not the Watch.