You have \$N=2\$ switches (inputs), meaning that there are \$2^N=4\$ possible combinations of switch positions.
Before proceeding to tinker, it's best to formally define the lamp states that would result for each input condition, in a table:
Switch 1 |
Switch 2 |
Lamp 1 |
Lamp 2 |
Open |
Open |
|
|
Open |
Closed |
|
|
Closed |
Open |
|
|
Closed |
Closed |
|
|
Fill in columns 3 and 4, and then you can proceed.
As it stands, we are being asked to guess what "one lamp disables the other" actually means. This implies some symmetry to the problem, but if you think about it, one of the lamps will need priority in the case where both switches are on. Otherwise the system would have to have some form of "memory" of prior states. This suggests a fundamental asymmetry entirely absent from your description of it, which would become immediately clear if you formally described behaviour in the truth table.
What also becomes clear when you write out a truth table, is if the problem even has a solution using simple passive and combinatorial elements. It's unlikely to be an issue in this case, but the argument for a formal statement of input vs. output requirements is still strong. By forcing yourself to imagine the state of affairs for every single input condition, you must necessarily become aware of practical logical flaws.
Also, you say you want to use two batteries, implying that there is some independence between each set of "battery+switch+lamp", but almost every case in that table is possible to implement with a single voltage source. It's therefore also necessary for you to explain to us the reason for having two independent power sources. Without this information it is impossible to guess how they should interact, whether it's OK for one single battery to power both lamps, and so on.
You have not stated if it's OK to use relays, or transistors in the solution. Perhaps you wish to use purely passive elements, namely switches, lamps and batteries, or perhaps you are willing to consider active parts.
In short, your question is vague, and can't be answered in its current state.