Sharing 1 dryer outlet
The shared washer-dryer is a commons-space load.  Those are required by various Codes to be on their own meter.  This is obvious in an apartment building with parking lot, hall and stairway lighting, coin operated laundry etc.  Duplexes are often built as independent homes with no commons spaces at all, and I would imagine that's what you had if your AHJ permitted your build with 2 meters.  Unless the tenants agree to do it on a handshake (that's what we did), you'll need to go back to the AHJ and have a conversation about this sutuation, and whether a changeover switch would be allowed.   

Generally with a washer-dryer, you have 6 utilities to plumb. 


Cold water
Hot water
Sewer
120V (low power) electricity
Gas or 240V (high power) electricity
Dryer vent


So it's not quote as simple as a transfer switch.  

Just swap plugs

Were it any other load, I would propose blue receptacles and red receptacles, and have the tenant switch the cords.  However the large dryer receptacles are not durable enough for frequent change.  Breaking a neutral on a NEMA 10 plug or socket will kill your tenant, but you will hook it up NEMA 14.  They do make frequent-swap styles, such as NEMA L14-30 or other makes; the solution could be as simple as that.  

Subpanel

Since you need to power the washer too, and possibly a water heater, a subpanel may be called for. This might seem like a clever way to kill 2 birds with one stone using "generator interlocks" made for a panel: wrong!  This application requires we switch neutral also, which calls for a 3-pole  transfer switch.  You could do that with a 3-phase panel and wire neutral as a phase, with two breakers backfed to the two supplies, and the dryer and washer on additional 3-pole breakers.  It would be up to the AHJ to understand and approve that.   Otherwise you'd need an external transfer switch with 3 poles, and rated for the current.  That's one neat thing about subpanels as transfer switches, they can handle high current cheaply, since they are just breakers.  

Each tenant would need to shut off their house breaker to keep the other tenant from just leaving the switch on them and poaching power. That might be a problem for the AHJ.  A better answer might be an Intermatic style timer rundown switch that operates a contactor, so the tenant can set the timer and power the washroom for 2 hours, say.  The contactors alone could not be the interlock, unless the AHJ certified that, and I doubt they would.  

60A is a good feed current as it works with #6 wire, the breakers are still cheap, and will power an electric dryer (23A), a washer (12A on half the circuit) and a "20A" on demand heater (15A) without kissing circuit limits too badly.