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As noted in 6v6gt's answer, if the ATmega328 has been set to use a crystal, thereafter a crystal (or an external oscillator) are required for programming, until the fuses have been set to use the internal RC oscillator. Here are a few thoughts, unfortunately somewhat scattered, on the matter.

If you have a parallel programmer (that delivers high voltage to the Reset pin and uses 8-bit parallel data signalling) you can use it to reset the fuses, since with that programming mode the clock is irrelevant.

Also, I think some versions of USBasp generate a programming clock and may be able to reprogram fuses away from the crystal setting. However, some USBasp clones ignore AVRdude's commands for frequency setting.

Rather than "sharing" a crystal via wires from one board to another – a practice that will broadcast lots of RF noise and might not get the signal across anyway – you could instead write a program on one Arduino to toggle a pin at high frequency, and use that output as an external oscillator to drive one of the crystal pins on another chip. For example, the program below should generate a 2.67 MHz square wave on pin PB0 of an Uno or a Nano.

void setup() {
 pinMode (8, OUTPUT); // PB0 is pin 8 on Uno / Nano
 while (1) PINB = 1;
}
void loop() {}

As noted in 6v6gt's answer, if the ATmega328 has been set to use a crystal, thereafter a crystal (or an external oscillator) are required for programming, until the fuses have been set to use the internal RC oscillator. Here are a few thoughts, unfortunately somewhat scattered, on the matter.

If you have a parallel programmer (that delivers high voltage to the Reset pin and uses 8-bit parallel data signalling) you can use it to reset the fuses, since with that programming mode the clock is irrelevant.

Also, USBasp programmers interact with Avrdude to generate a programming clock of some specified frequency, 100kHz by default, 8kHz with the "Slow Clock" jumper installed. Also see § "Set the clock in software" within USBasp ICSP programmer quickstart guide at freetronics.com.au. Unlike the Avrdude man page, that reference says the default programming frequency is 375kHz and says "Avrdude’s -B option allows you to set the programming clock speed from software", while the man page says -B specifies bit clock period (in microseconds) for the JTAG interface or the ISP clock of the JTAG ICE only. Anyhow, a USBasp can reprogram fuses away from the crystal setting if a crystal or external oscillator is used, without depending on serial port data rates. I don't know for certain whether it will work without a crystal or oscillator.

Note, rather than "sharing" a crystal via wires from one board to another – a practice that will broadcast lots of RF noise and might not get the signal across anyway – you could instead write a program on one Arduino to toggle a pin at high frequency, and use that output as an external oscillator to drive one of the crystal pins on another chip. For example, the program below should generate a 2.67 MHz square wave on pin PB0 of an Uno or a Nano.

void setup() {
 cli(); // Avoid timer interrupts
 pinMode (8, OUTPUT); // PB0 is pin 8 on Uno / Nano
 while (1) PINB = 1;
}
void loop() {}

In the ATmega328 data sheet, see §14.3.1, Alternate Functions of Port B, and Table 14-3, which lists PB7 as XTAL2 (Chip Clock Oscillator pin 2), and PB6 as XTAL1 (Chip Clock Oscillator pin 1 or External clock input), indicating external oscillator attaches to the PB6 pin.