Hi
There are two different versions of Apple Watch Series 3:
All models of Apple Watch can be used to send and receive texts and to make and receive calls when your iPhone is nearby and connected to a cellular network and, potentially, also under other certain other circumstances (see below):
Apple Watch Series 3 (GPS)
Apple Watch Series 3 (GPS) models do not have a built-in cellular capability.
Any calls or data-related services are routed either via the paired iPhone or via a direct connection between the watch and a compatible Wi-Fi network (when the watch cannot connect to the paired iPhone via Bluetooth).
A separate cellular plan for the Apple Watch is not required.
Apple Watch Series 3 (GPS) can be used in conjunction with an iPhone 5s or newer model of iPhone (running the latest version of iOS) and a service plan for the iPhone from any carrier.
Apple Watch Series 3 (GPS + Cellular)
In respect of data usage, Apple Watch Series 3 (GPS + Cellular) also always first tries to use the Bluetooth connection to your iPhone (which does not require a separate plan for the watch) or, in the absence of that, any available connection to a compatible Wi-Fi network.
If neither of those is available, then Apple Watch Series 3 (GPS + Cellular) has a built-in cellular capability that, under certain circumstances, can enable certain features, which are not available on GPS-only models, to remain available when the watch is used without the iPhone nearby.
The feature involves the built-in eSIM / embedded SIM card in the Apple Watch copying information from the paired iPhone's SIM card, including using the same phone number.
The feature requires a carrier plan for your iPhone that specifically includes support for the Apple Watch Series 3 (GPS + Cellular), including the carrier activating the device on the plan:
An activation fee and ongoing monthly cost may apply - contact your carrier for details:
Apple Watch - Carriers - Apple
Apple Watch Series 3 (GPS + Cellular) can also be used without enabling the built-in cellular feature, in which case it functions like a GPS-only model and can be used in conjunction with a service plan for iPhone 6 or newer model of iPhone, running the latest version of iOS, from any carrier, with no need to activate the watch on the plan.
All models
The functions that are available on your Apple Watch when it is away from your iPhone depend on various factors including:
The features and requirements that apply under various circumstances are detailed here, including within footnote #1 in respect of text messages:
How to use your Apple Watch without your iPhone nearby - Apple Support