You want telepathy with non-telepaths? .. and a plausible anatomy in the telepath to explain that?
Sorry, but you can't have it.
The nearest thing would perhaps be an organ that replicates a crystal radio set, and that requires those it's 'telepathically' 'talking to' to also have radios to 'hear' anything said to them .. so .. within the bounds of real science what you're asking for is impossible .. while without the bounds of real science you're just using magic and 'plausible' anatomy becomes irrelevant.
Alternative potential answer
If we consider the anatomy of the 'non-telepaths' as well as your telepaths.
A potential answer does present itself.
All members of the species have organic crystal radio sets as part of their anatomy.
Only some (those considered to be 'telepaths') have any voluntary control of them however.
If you want your telepaths to be able to read non-telepaths minds then the non-telepaths are involuntarily and subconsciously constantly broadcasting their thoughts.
Which means that, unless you want everyone hearing everything both telepaths and non-telepaths broadcast, that your non-telepaths must somehow only 'hear' anything that's being broadcast if it's directly addressed to or at them by a telepath .. or maybe their sets only send and can't receive, so your telepaths can't talk to them and can only read their minds.
Alternatively a telepath that connects to ('dials?') a 'non-telepath' can choose to put his 'radio' on mute and just listen in to his targets thoughts .. so their connection is like a mobile phone call.
In short, think of it like this .. everyone has a cell phone in their head, most people haven't any credit on theirs so can only receive calls .. and most peoples phones are infected with a virus that turns them into a bugging device .. likely range will be around the same as children's toy walkie talkies as you've no cell phone towers to boost signals on, so perhaps around a mile.
None of the members of the species need to know any of this, neither do any readers (or players?), It can just be for the authors use to help rationalise how it works and formulate rules for what can and can't be done with it.
"None of the members of the species need to know any of this" in fact it's probably best if they don't, or at least that it's not common knowledge .. because the obvious way to prevent your mind being read is to just have the organ surgically removed and you probably don't want that .. 🤔 or wear a faraday cage tin foil hat, they'd really work as advertised for a species like this 😁