Low observable spacecraft were conceptualized by the United States and reputedly one code named "Misty" was launched in the 1990's. This satellite used "balloon" technology to envelope the satellite in a cone shaped cover which deflected radar and diffused light to the point the satellite was almost invisible without advanced tracking gear (presumably infra red or thermal imaging.
Today, we are on the verge of being able to use metamaterials to refract light or electromagnetic radiation around the satellite in paths the designers choose, effectively allowing the radiation to "flow around" the spacecraft so there is no "hole" to alert an observer. Misty might become noticeable by occulting a star or planet as it passes in front of the object, but a metamaterial shielded object will not. Obviously there will need to be "holes" in the shield for sensors to operate, and keeping the spacecraft safe from observation in thermal imagery will be extremely difficult, since the spacecraft is operating against a 3K background, and the power sources and any rocket burns will stand out as bright beacons against this.
The X-37 program shows a way to enhance this, the spacecraft is designed for long duration flight and has the ability to change orbits. If the stealth satellite can make orbital manoeuvres over friendly territory or the mid Pacific where it is less likely to be seen by ground observers, and potentially every orbit can be different, making tracking much more difficult.
As for the second part of the question, if the ground station is using tight beam communications, it will be impossible to intercept from the ground. To make it even more difficult, the ground station might send a tight beam to a relay satellite, so even the orbital parameters of the stealth satellite cannot be determined by observing the angle the ground dish is pointed at.
The only plausible way to crack this problem is to either be on the inside at the ground station in person, or to have inserted some sort of tracking program which records the transmissions and downloads to and from the satellite. Even if it is heavily encrypted, the pattern of transmissions and receptions to and from space with no corresponding satellite would be very illuminating, and alert an enemy intelligence service that something is in orbit.