What is the verb for "accidently hearing"?

Cambridge dictionary defines overhear as

to hear what other people are saying without intending to and without their knowledge

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Oxford dictionary defines the word as

Hear (someone or something) without meaning to or without the knowledge of the speaker.

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Lots of times you "overhear" someone by accident and then it's your choice whether to keep listening intentionally or not to do so. When you begin to intentionally listen you are not really "overhearing" someone or something, rather you're deliberately listening to someone, probably without their knowledge–and that becomes eavesdropping.

You've shown that the American Heritage dictionary does not mention anything about the hearer's intentions but the two dictionaries I cite do so, with Cambridge including the 'accidental part" in the meaning. Two or three other dictionaries–including the Oxford English Dictionary, Merriam-Webster, Collins–do not mention the hearer's intention, but this is not as full a definition as the two I've included, and with which other speakers agree in comments.

I agree that overhear does not imply that you were intentionally listening, and can work for your situation. However, if you want to stress both your inability not to listen and the fact that the speakers didn't care whether you heard or not, you could use the phrase subjected to. From Merriam-Webster:

to subject (someone or something) to

1 : to cause or force (someone or something) to experience (something harmful, unpleasant, etc.)

This phrase isn't limited to sounds, so you would need to specify what you were subjected to; for instance

Walking down the street I was subjected to the loud argument/obscene language/private conversation of some people on the corner.

Also note that it needs to be subjected to; plain subject to means something slightly different.