It looks like the answer is yes. Recently it occurred to me that it was very possible Siri heard us talking about product or other we needed. Or, some work that needed to be done around the house. The reason I thought this was because within a day of those conversations, I'd see prominent ads on my Facebook page, or Google search page or an email promoting that very same thing. At first I just thought it was coincidence that an ad front and center for a certain cleaning product hit just as I went out to get it. The movie we saw last week was the latest bit of coincidence. Dale and I talked about going to see The Intern and suddenly there was a ad for it on my page.
Listening devices are on phones, Google Now, Microsoft's Cortana, and Amazon's Echo. I have Siri and the Echo. I like the Echo tower which just sits on our dining room table waiting for me to ask it a question or to turn on KMOX. But all this is kind of spooky if we're honest about it.
There is some evidence that Siri and other verbal command personal assistants on devices may in fact be listening and sending off that info to sellers. They are always listening, waiting for a command or question from you. This shouldn't be too surprising since we've learned that carrying a phone through a store brings on the good possibility that your shopping is being tracked.
The best advice if you aren't comfortable with all this is to turn off the device when you don't need it.