Assuming the LEDs are multiplexed, you can use a single resistor if only one LED will be on at once (instantaneously on, that is).
If the ground symbol is replaced by a switch you can multiplex n x 4 LEDs with 4 resistors, where a practical value for n might be something like 4 or 8.
In general if you want to use persistence of vision to have the display have any combination of LEDs appear to be on simultaneously then you will use an n x m matrix and you will need either n or m resistors. Limitations on peak current for LEDs mean that you will not generally want to make an n x 1 matrix where n is larger than 8 or 10 (the peak to average ratio will be too large for a bright enough display with a given LED).
It is conceivable that you could use matched crummy drivers (essentially acting as ballast resistors) and change the duty cycle to equalize the apparent brightness with your 2nd circuit, but that would be complex and might not match all that well (and resistors are really, really cheap).