Yes, most modern computer processors include hardware with the capability to fully control all components of the computer (regardless of the power state of the system as a whole), to access all data while the computer is running, and to connect to the internet (in any power state).
However, the remote control aspect of the functionality this hardware provides is not enabled on most devices targeted at the consumer market.
Intel Management Engine (and similar systems)
The Intel Management Engine, referenced in the question, is present in almost all Intel chips sold since 2006. It is an independent computing environment, which has access to (and control over) the main processor, the memory, the network interfaces, and other systems.
One of the primary purposes of the ME is security: it verifies the integrity of the firmware running on the processor and on the Trusted Platform Module.
Additionally, the ME enables a remote management system for enterprise use, called AMT (see below). Most consumer devices ship with this functionality disabled in the firmware.
AMD has a similar system called PSP.
Remote management (AMT etc)
One of the services provided by the Intel ME is called Intel Active Management Technology. AMT enables "lights-out management", meaning it enables system administrators to remotely control and modify virtually all aspects of the system, including the ability to download and update software and firmware regardless of whether the computer's operating system is running. (Obviously the battery or power supply has to be connected.)
This type of remote management originated in servers, where it originally used a dedicated network interface. However AMT uses the system's normal built-in networking interfaces including ethernet, wifi, and (in rare cases) 3G.
AMT is part of Intel's "vPro" technology, which is found in a wide variety of devices. It is primarily targeted at enterprise users, however it has made its way into may devices available on the consumer market including laptops primarily targeted at business use, as well as high-end gaming hardware.
The AMT system is normally not enabled on computers targeted at the consumer market; however the hardware is still there and the Intel Management Engine is still active because it provides other functionality too (see above).
It is important to note that the one of the main purposes of the Intel Management Engine and similar technology is to increase security.
Because it verifies the integrity of the firmware running on the processor and other vital system components, it ensures that this firmware has not been modified or replaced with potentially malicious versions. (Or any other modifications – it simply ensures that only the original firmware can be used.)
However, the Management Engine itself is not entirely immune to compromise.
In the past researchers have been able to remotely compromise the system and gain control of machines without physical access to them.
Another concern (more relevant to high-risk users such as non-US governments and political dissidents) is that technically there is no reason why the Management Engine (or similar components in other chips) couldn't contain backdoors allowing government agencies the same access and control over the system.
Intel is a US company (though a significant part of their engineering is based in Israel), and they could be required by US government to implement hidden backdoors.
Since it is impossible to audit the firmware, no proof either way is possible as to whether backdoors exist or whether the risk is purely theoretical.
This Hackaday article is informative, if somewhat hysterical, look at the features and security risks of the Management Engine. (Thanks to @William-remote for sharing it in a comment on his answer below.)
See Intel's page on AMT.
In the past Intel provided an anti-theft service to enterprise and consumer markets, whereby the ME would regularly check ion with Intel servers and disable the computer if it had been reported stolen. Intel have now discontinued this service.
An HP document on the use of AMT (thanks to Igor Skochinsky for sharing in his answer below).
There is a generic set of standards for a functionality similar to AMT, called IPMI.
I hadn't expected to answer this question myself, but having done some research I felt I was in a position to do so.
Thanks to everyone who has contributed with additional information. I will continue to incorporate any new information I find into this answer.