Erdin:
Yes, there will be a current of 1mA, but nothing is damaged.
When an I2C device is communicating, it also lowers the I2C signals to 0V now and then. That is how the I2C works with the pull-up resistors. The resistors make the signal high, and the Master and Slave pull the signals low to communicate.
The I2C bus will no longer work when an Arduino is pulling the I2C bus down.
And 1mA might in some cases be enough to make the Arduino start working a little and even make a power led to light a little.
I connected one Arduino Nano and Arduino Uno together with 1 ground wire, 1 SDA with 4.7 K ohm resistor pullup to 5V of Nano, and 1 SCL with 4.7 K ohm resistor pull up to 5 V of Nano. Then 1 usb cable from Nano to an usb-hub of a laptop, and 1 usb cable from Uno to the same laptop hub.
Test examples of Arduino ide worked well with 100 kHz speed and 400 kHz speed, although windows XP did not always recognize the usb devices at the first usb-connection attempt. (I used a separate usb-hub without external power supply to connect the Arduinos to the laptop).
The 5V pin in Nano had 4.54 V (!) and 5 V pin in Uno had 4.86 V.
When usb cable from Nano was disconnected, the 5 V pin in Nano still had 0.22 V.
When Nano was reconnected with the usb, but Uno disconnected, the Uno 5V voltage was 0.86 V.
When the pull up resistors were moved from the Nano power supply to the Uno power supply 5V, the unpowere voltages changed: Uno 0.19V, Nano 0.89V.
It is quite comfortable to be able to start two or more Arduino ide processes with their own boards and serial ports in the same computer and see the monitor windows from all Arduinos side by side!
But I am still concerned whether it is safe enough for me to let 1 milliampere current flow to an unpowered Arduino through the SDA pin internal protection diode(?) and another milliampere through SCL internal protection diode. There are strong opinions in the web that internal protection diodes should be allowed to have only microampere currents at most as designed. E.g. Russel McMahon's current table here.
Would it be good to add 8 external protection diodes:
SDA Uno to ground
SDA Uno to 5V
SCL Uno to ground
SCL Uno to 5V
SDA Nano to ground
SDA Nano to 5V
SCL Nano to ground
SCL Nano to 5V
e.g. Schottky 1N5819 ?
And perhaps increas the pull up resistors from 4.7K to a double or tripple value to decrease the current to the unpowered Arduino?