They are legal and reasonable requests. Put yourself in their shoes, would you want random people wandering through your home when you were not there? You lock your doors at night and when you leave because you do not want random people coming into your home. To a tenant, these people you are showing the house to are complete strangers with unknown criminal histories that could be potential thiefs.
When you rent a property, you give up your right to control access to the property during the tenancy. The tenant has superior right of access to the owner of the property, that is what the tenant is paying rent for. RCW 59.18 contains a section stating that the tenant shall not unreasonably withhold access to the property if the landlord provides proper written notice at least 24 hours in advance of showing the property and 48 hours in advance for other issues. It is very reasonable for the tenant to request to be around when strangers are wandering through their home. The tenants do not seem to be withholding access, but rather asking to have their privacy respected.
Working with your tenant to come up with a reasonable plan and times to show the rental may help alleviate the tenants concerns and provide you more opportunities to show the rental. Taking five minutes to understand the tenants concerns will take you a lot further than attempting to force your will upon them.
This answer is for informational purposes only and should not be construed to establish an attorney client relationship. Before taking any legal action, it is always advisable to discuss your specific situation with an attorney.