I'm going to join the people saying "no" here.
There are actually quite a few circuits that carefully isolate one part from another.
Transformers
From the planet Cybertron...oops, no, not them. The old standby is a transformer. A typical transformer has a core (typically thin sheets of iron, but sometimes other stuff like ferrite) with some wire wound around that's connected to one circuit, and another wire wound around it that's part of another circuit. The two "windings" (pieces of wire) are insulated from each other though.
The "primary" winding is driven by an incoming signal. It forms a magnetic field in the core. That magnetic field in the core then drives current in the secondary winding. So, we can transfer power across, but there's no electrical connection between the two at all. Transformers are routinely used both for transferring power, and transferring communication signals.
In a fair number of cases, a secondary winding will have a center tap that's often connected as the ground in the circuit powered by the transformer. The two "outer" wires from the secondary aren't connected directly to the primary, but they at least sort of correspond to the two inputs to the primary. The center tap, on the other hand, doesn't even correspond to any input.
Opto-isolators
When you're dealing solely with communication (not power), it's also pretty common to use optoisolators. An optoisolator is basically just an LED next to a phototransistor. An incoming signal turns the LED on and off. The phototransistor senses when the LED is on or off, and controls current flow based on it. The result is basically the same: we allow communication between two circuits, even though we isolate them, so there's no actual electrical connection between them at all.
Capacitors
Yet another form (so common many are likely to almost forget them) is the simple capacitor. A capacitor is simply two conductors separated by a (usually quite thin) insulator. An incoming signal causes a static charge on one plate, which causes a static charge on the plate next to it, so the output tracks (to some degree) with the signal on the input--but (again) from a viewpoint of sharing a common ground (or anything similar), the two are isolated from each other.
Summary
Although it is pretty common for circuits to have a common ground, it's also fairly common to completely or partially isolate circuits from each other in various ways that allow communication between them, without any common reference point or any possibility (short of a short in the circuit board, or something on that order) of actual current flow between the isolated sections.