1. What does it mean?
갠톡 is a compressed form of 개인톡: a private, individual chat rather than a group conversation. It is commonly used when moving a discussion out of a group chat.
Also written or said as 개인톡, 갠톡하다, direct message
2. The nuance
'Private' here means one-to-one, not a promise of technical secrecy. In 갠톡으로 보내 줘, the speaker is asking you to send the message directly. On another service, 디엠 or 개인 메시지 may be a more natural label.
3. Examples
“주소는 단체방 말고 갠톡으로 보내 줄게.”
I'll send the address in a direct chat instead of the group chat.
“일정이 안 맞는 사람은 나한테 갠톡해 줘.”
If the schedule doesn't work for you, message me individually.
Sources
National Institute of Korean Language
References support the meaning and usage. Example sentences are written for NoCapWiki.