No. They are often left off, if the context makes it clear who is talking. If I only have Mike and Nancy in a scene:
"I had ice cream at lunch," Mike said.
"I thought we agreed we would have it together?"
"I just forgot."
"Okay," Nancy said, "Thanks for thinking about me."
Typically don't go too far without attributions, no more than two or three lines of dialogue. Even with just two characters speaking, a reader can get confused as to who is speaking, and you do not want them to have to stop and count, like "this is Alan, then Betty, then Alan, then Betty, ... oh this is Alan saying 'No' and Betty says 'Yes you will' ..."
Another way to make attributions is by camera focus. If a character does something, and speech follows, the reader assumes that character spoke.
Betty rolled her eyes. "You have to be kidding me."
John stopped at the top of the stairs and stopped to lean on the wall and catch a breath. "When I was a kid I'd run up these stairs three at a time."
That said, readers are so accustomed to "Alan said", "Betty said", "Charlie said" that these register to anchor speech to character nearly subconsciously, the text does not bother them. So you should always put it anywhere the speaker isn't clear, it isn't going to interrupt your reader's attention or imagery. If there are more than two speakers in the scene, that is on every utterance.
If you have just one person speaking, like telling a story, you can break that into paragraphs without any "she continued" or "she went on" or anything at all.
Just leave the closing quote off of a leading paragraph, and add a new opening quote on the next, and that means it is the same speaker. Do the same thing to start another paragraph with the same speaker. Add the closing quote to the final paragraph.