My favorite educator, Bill Beaty, often rants at the many misconceptions that all too many people have been infected with.
One of the many common misconceptions involves batteries.
"Frequently-Asked Electricity Questions":
"THE LIQUID BETWEEN A BATTERY'S PLATES IS A GOOD CONDUCTOR.
SO WHY DOESN'T IT SHORT OUT THE BATTERY?"
"Why is electricity so hard to understand?"
"...mistaken belief that no charge flows through batteries. ...
This leads to the traditional incorrect flashlight-current explanation (current comes out of battery, flows...etc.)
It also leads to the misconception that batteries
SUPPLY CHARGE, and have a storage place for "used" charge.
This might make sense if we believe that there's no path for charge through the
battery.
But it's wrong, because there is a path, a path provided by
flowing charged atoms.
Charge must flow around and around a circuit,
passing THROUGH the battery over and over."
"But how SHOULD we teach kids about 'electricity'?"
"A battery is a chemically-fueled charge pump. Like any other pump, a battery takes charges in through one connection and spits them out through the other. A battery is not a source of the "stuff" being pumped. When a battery runs down, it's because its chemical fuel is exhausted, not because any charges have been lost. ...
When you "recharge" a battery, you are pumping charges through it backwards, which reverses the chemical reactions and converts the waste products back again into chemical fuel."
'Which way does the "electricity" really flow?'
"When you connect a lightbulb to a battery, you form a complete circuit, and the path of the flowing charge is through the inside of the battery, as well as through the light bulb filament. Battery electrolyte is very conductive."
Where is that quote from? It is self-contradictory, and pretty much just plain wrong.
Current will flow if a load is connected between a generic power supply positive and negative, no ground connection is needed, unless you have some special purpose power supply.
A ground connection is often used for safety, especially in line non-isolated supplies, or as a means to reduce noise. Even so, not all supplies pass the ground on to the output terminals or even make it available (for common examples, think wall-warts and such with just a two pole output). Many supplies don't even have an incoming ground terminal from the mains (small switchers, and again, wall-warts).
I was going to post this as a comment, but it seemed to cover the question.
Edit as per comment below and corrected text.
Well, that does change things. Generally, a DC power supply has no output tied to ground, or connected to the supply electrically at all, so connecting a load between an output and earth ground would result in no current flow. Supplies that have an output set of terminals and a separate ground can often be configured (by jumping ground to one of the output terminals) as positive ground or negative ground if desired.
Best Answer
It is implied that the 9 V source has its negative end connected to ground. Current therefore flows out the positive end of the 9 V source, into the circuit, out the circuit's ground connection, and back into the negative end of the 9 V source.
Added
As Dave Tweed pointed out in a comment, and I didn't notice ealier, the second circuit uses a positive ground. In that case, current flows out the positive end of the 9 V supply into the circuit ground, thru the circuit to the -9V connection, and back into the negative input of the 9 V source.
The common thread in both cases is that ground is implied as the 0 V reference for all other voltages that don't explicitly specify a reference. Both the "+9v DC" point in the top circuit and the "-9v" point in the bottom circuit are therefore relative to ground. Since they are labeled as supply points, it is assumed that a power supply is connected between these points and ground.