NEC 2008
110.14 Electrical Connections. (A) Terminals. Connection of conductors to terminal parts shall ensure a thoroughly good connection
without damaging the conductors and shall be made by means of pressure
connectors (including set-screw type), solder lugs, or splices to
flexible leads. Connection by means of wire-binding screws or studs
and nuts that have upturned lugs or the equivalent shall be permitted
for 10 AWG or smaller conductors. Terminals for more than one
conductor and terminals used to connect aluminum shall be so
identified.
So if more than one conductor can be placed under a lug, it must say so somewhere on the panel (or in the panel documentation).
408.3 Support and Arrangement of Busbars and Conductors. (D) Terminals. In switchboards and panelboards, load terminals for field
wiring, including grounded circuit conductor load terminals and
connections to the equipment grounding conductor bus for load
equipment grounding conductors, shall be so located that it is not
necessary to reach across or beyond an uninsulated ungrounded line bus
in order to make connections.

If you have a bus bar on each side of the panel like in the diagram above, you should not connect the neutral to one bus and the ground to the other. Neutral and ground from all circuits on the left should go to the left bus bar, and Neutral and ground conductors from the right should go to the right bus bar.
408.41 Grounded Conductor Terminations. Each grounded conductor shall terminate within the panelboard in an individual terminal that
is not also used for another conductor.
Exception: Grounded conductors of circuits with parallel conductors shall be permitted to terminate in a single terminal if the terminal
is identified for connection of more than one conductor.
So if the panel allows multiple conductors under a lug, you can terminate multiple equipment ground conductors under a single lug. However, you cannot use a single lug for multiple grounded conductors (neutrals), or a mix of equipment ground and grounded (neutral) conductors.
Here's why you can't have multiple neutrals in a single terminal.
Multiple neutral conductors in a single termination create a
significant problem when the circuit needs to be isolated. In order to
isolate the circuit, the branch breaker is turned off and the neutral
is disconnected by removing it from the terminal. If the terminal is
shared with another circuit, the connection on the other (still
energized) circuit will be loosened as well. Loosening of the second
neutral (loss of neutral) under load is a safety hazard, and may
establish an overvoltage condition on lighting and appliances if the
neutral is part of a 120/240 Vac multi-wire branch circuit. Source
And this is why you can't have a neutral and a ground in a single terminal.
The connection of a neutral and equipment-grounding conductor in the
same termination creates a similar issue. One of the objectives of the
particular arrangement of bonding jumpers, neutrals and equipment
grounds is to allow circuit isolation while keeping the equipment
grounding conductor still connected to the grounding electrode (see
UL 869A - Reference Standard for Service Equipment). When the neutral
is disconnected, the objective is to still have the equipment ground
solidly connected to the grounding electrode. If both the neutral and
grounded conductor is under the same terminal, this cannot be
accomplished.
Source
You can , however, have both grounded conductors (neutrals) and equipment ground conductors connected to the same bus bar in the main service panel if the grounded conductor bar/bus (neutral bar) is bonded to the equipment ground bar/bus (it's different if you are dealing with a sub-panel, since the bar/bus will not be bonded).
Long story short
You'll have to check the panel documentation to determine if multiple conductors can terminate under a single lug. If they can. You can connect a couple equipment ground conductors to a common terminal, which should free up enough space to add the breaker.
NOTE: This only applies to bus bar terminals, most breakers are not rated to be "double tapped". So you should never have two conductors under a single breaker lug.
Absolutely, you can run a separate ground - they have allowed this for a long time and have recently liberalized the rules on retrofitting grounds to practically everything else. So if you have other outlets that could also use a ground, food for thought.
The grounds do not need to follow the same route as the original wiring. They do need to be thick enough for their purpose. Also, other circuits can share a ground wire, as long as they originate from the same panel.
Should you do it? Absolutely! The problem is, right now you have the NEMA 10 problem. Normally if a neutral wire breaks, the neutral wire is pulled up toward 120V because power has nowhere to return, but we don't care because the grounding system still protects us. Now, if ground is also the same wire (like in NEMA 10), it too will be pulled up toward 120V - and now things which are supposed to be grounded will instead shock you! Including switch plate screws, metal chassis of equipment, and of course, the metal chassis of your subpanel, which you will be opening directly when you realize there's a problem. Touch that and a water pipe, and blammo!
Best Answer
If you want two breakers, you can't share a neutral and ground. The shared neutral and ground could be exposed to too much current, because they don't have breakers for protection, only the hot does. An example:
Assuming you've got an existing circuit with a 15 amp breaker and 14 gauge wire, you add a second 15 amp breaker, running a new 14 ga hot, and sharing the neutral and ground from the existing circuit.
You have one load plugged into the "old" outlet, drawing 10 amps. 10 amps comes out through your old breaker and your old hot, the old 15 amp breaker sees 10 amps and doesn't trip, and the 10 amps goes back through your old neutral.
Now you plug in another load to your "new" outlet, drawing 12 amps. 12 amps comes out through your new breaker and your new hot. The new 15 amp breaker sees 12 amps and doesn't trip. 12 amps goes back through your old neutral. Now you've got a problem.
Your old 14 gauge neutral is now carrying 10 amps from the old circuit + 12 amps from the new circuit = 22 amps: thats more than the rated capacty. You haven't tripped a breaker, but you've overloaded your wiring. This is a very dangerous situation.