Nanny here I use both.
I use written letters as part of my portfolio and during the application process. I use it as one of my tools to introduce myself to a family.
I have permission to share my previous employers phone numbers, but I don't share that information with everyone, only if we're at a certain point in the process.
Honestly, I want a potential employer to call my references and chat. I'd be concerned if they didn't want to speak with them. To me it's a sign they are responsible and thorough.
MB here. I want to speak directly with former employers. I don't completely disregard letters of reference but I don't consider them in any way sufficient.
There is no substitute for talking directly with another mother in my opinion.
I've also written letters of reference and what I don't say in them is as important as what I do say. So a prospective employer who calls me for a reference will get a great deal more information than could ever be put in a letter. Also, the myriad things that influence "fit" cannot be captured in a reference letter. Someone else's nanny who wasn't a great fit might be perfect for us, and vice versa, but you can only discern information like that from a conversation (assuming you ask the right questions).