NASA and the ESA have Deep Space Networks that circle the globe enabling them to track their spacecraft constantly. However, Russia, China, India, and possibly also Japan, only list tracking stations within their national borders. While China is apparently planning on setting up an additional station in South America, how do these countries currently track their spacecraft when they are not in sight? Do they rely on the NASA or ESA networks?
Unless there is direct mission cooperation with NASA or ESA, these countries generally do not rely on help from those two organizations. Exceptions to this are when there is a failure or other spacecraft emergency, as with the recent Phobos-Grunt mission.
What do they do instead? Well, if we're just talking tracking, there's usually no need to constantly track a spacecraft. A few passes per day will give you enough info to nail down your orbit, and in LEO you will get at least a couple passes per day over your home country. Of course, these countries also have at least some semblance of a GNS system, which can also be used.
In terms of data relay, often strategically-placed relay satellites help solve this problem. For instance, Russia has its own Satellite Data Relay Network, similar to the US's TDRSS, which can receive transmissions from a satellite in orbit and send the signal to a ground station.
One more thing: NASA's Deep Space Network is normally only used for very distant interplanetary missions. Significantly smaller ground station antennae can be used for Earth-orbiting satellites. For non-ESA/US countries, the same principle applies for interplanetary missions... just track it when you can see it.
To expand a bit more... yes, China and India appear to have data relay satellite systems, although if you're interested in interplanetary mission communication these would not be used. Instead, I just discovered that Russia, China, and India all have their own DSN's (which I now see you've linked to in your OP, oops).
Talking about India It has one DSN station with S/X band support with 11,18 and 32m antennas. As earlier answer marked, except for time critical tracking and control operations, data set collected over series of passes or a larger time interval can suffice orbit determination needs.
During critical mission phases of important missions like MOM, ISRO does get help from NASA DSN under conditions unknown. MOM was tracked simultaneously by Canberra or Goldstone station during the Mars insertion phase.
ISRO also used a set of Ship-borne Terminals deployed in pacific for tracking the MOM until upper stage ignition. Other TTC operations are coordinated by ISTRAC ground station network spread in and around India to track Earth orbiting spacecrafts and geo-centric phases of planetary spacecrafts.
Probably GPS-like system, swarm of satellites solve this task easily. See https://en.wikipedia.org/wiki/GLONASS