If they are rayon, getting wet can be a problem with older rayons. If they are nylon, or cotton or a synthetic/cotton blend, probably fine. Silk, hmm, it depends somewhat, some silks handwash beautifully, others need to be dry cleaned.
That said, I have a 1960s satin quilted robe, made of nylon I think, looks similar to your right-hand one, that I've been putting through the machine now and then, on a delicate wash with gentle soap, and it's been fine.
If you want to take the risk, try a short cold hand wash with a gentle detergent. Rinse and gently squeeze out the water. Have some large dry towels to hand, lay the robes on them, and roll up the towel and squeeze to remove more excess water (the weight of the water can pull things out of shape if you drip-dry from very wet).
If they were mine, I would probably risk hand washing them, unless I thought they were rayon.