Yes, it is possible and it is the best way to communicate with KakaoTalk users.
You can send an AlimTalk or FriendTalk to the user and the user can directly reply to the same chat. When a user initiate a chat then all messages will be sent as ConsultationTalk.
Note:Alimtalk and FriendTalk are one-way outbound messages initiated by the business. While ConsultationTalk can only be inititated by the user. They are two different message types so you will need to enable Chat on the dashboard to get inbound messages.
The Sinch Developer Documentation site contains comprehensive documentation and tutorials, which show you how we support KakaoTalk using the Sinch Conversation API.