Are passengers more likely to see other airliners from a window seat on the right or on the left?
Your best bet is flying across the Atlantic, where the Strategic Lateral Offset Procedure (SLOP) is used.

This is where the pilot flies a offset to the route, either 1 or 2 NM, always to the right, chosen randomly (or based on surrounding traffic/TCAS), and is kept from the ATC (no radar coverage, although that is changing with satellite ADS-B).

As shown below, you have a chance to see another plane above/below from either side depending on the pilots' flip of coins.


Source; adapted

Regarding two-way airways elsewhere and with the navigation paradox, where flying accurately is ever increasing, it will be a slim chance except for a crossing traffic (not same airway).

If in a holding pattern, the standard turn is right turns, so the right side gives a better look "inside the track," unless the pattern is non-standard to the left.


Source: Mid-Atlantic Race!!! Virgin Atlantic A340 vs Norwegian 787 Dreamliner