Is it safe to touch wire testing at low to medium voltage on a 14 or 12 gauge wire, 15 or 20 amp (current) circuit?
No!
A non-contact tester is used to double-check that a circuit is on/wired correctly (i.e., if it doesn't light up when expected, then there is a problem) or off (i.e., if it lights up when you don't not expected, then there is a problem). When used to check that a circuit is off, that is typically to make sure that it is safe to work on the wiring:

Check circuit is on (using tester or by checking that light or appliance is on)
Turn off breaker or remove fuse that you think controls the circuit
Check circuit is off (using tester)

Remember, it takes a tiny fraction of an Ampere (GFCIs are rated in milliamps) at 120V to kill. SAFETY FIRST.
All that being said, there can be phantom voltage from another circuit or there could be low voltage (telephone, network, doorbell, thermostat) - neither of which would be a problem.
The ground wire (&quot;exposed copper wire&quot;) under normal circumstances will carry no current at all except when there is a fault in a circuit. There are limited circumstances (certain types of switches) where a very small amount of current is allowed on the ground wire. However, there are a number of other incorrectly wired situations that could cause dangerous current on the ground wire without any obvious fault, including:

Ground used instead of neutral - e.g., need two hots + neutral + ground, only have hot + neutral + ground, improperly used neutral wire as a hot and ground wire as neutral
Neutral and ground connected together where they shouldn't be, allowing all neutral current to flow on both neutral and ground
Wires simply connected to the wrong places and &quot;happen to work&quot;, but quite dangerous.

The only way to tell for sure is to check the voltage with a multimeter. The finger test is NOT RECOMMENDED.