do you need a separate cellular number and plan in order to use the watch without the iPhone?

Yes;

GPS

Apple Watch (GPS only) lets you send and receive text messages, answer phone calls and receive notifications when it’s connected to your iPhone via Bluetooth and Wi-Fi. The built-in GPS works independently of your iPhone for distance, pace and route mapping in workouts.

GPS + Cellular

Apple Watch (GPS + Cellular) has all the same features, as well as built-in cellular. It gives you the freedom to send and receive text messages, answer phone calls, receive notifications, and stream Apple Music and Apple Podcasts — even when you don’t have your iPhone with you.

What happens is you pay an extra monthly fee on your existing phone contract, the watch uses your existing phone number to make calls in the US it's about $10 per month and about £10 in the UK,that is for the cellular version of the watch, if you buy the GPS only version of the watch you can make calls and send texts as long as you have your phone with you,with the cellular version of the watch you can make calls and send texts without having your phone with you.So in reality your paying $10/£10 a month so that you can leave your phone at home.