The IoT is a whole indefinitely large number of things of different ways of connecting different things – is it could be fixed, Wi-Fi NFC, cellular, ultra-narrow band or even Zigbee. or have to mix and match what is best for each task.
From my humble point of view it can work with or without regular Internet. If, for example, you have a Wireless Sensor Network and it is isolated from any other agent, then it does not necessarily use the Internet, despite having their own routing network layer protocols (a loose equivalent would be an Intranet in an office).
However, if you get that WSN connected to a sink that is capable of behaving as a intermediary agent between the WSN an the "real" Internet, then you can retrieve readings and services from the motes composing that WSN by, for example, using Web services based on REST interfaces. You can get a more detailed explanation in this paper: http://www.mdpi.com/1424-8220/13/2/1787