Beth Chapman's Ashes Spread in Hawaii During Private Paddle-Out

There was a private and secret send-off for Beth Chapman in Hawaii that only Dog the Bounty Hunter and his family knew was happening ... TMZ has learned.

Sources directly connected to the family tell us ... this past weekend, Dog and his children spread some of Beth's ashes and flowers in Hawaii's Waikiki waters and bid her aloha in the state she loved and called home for many years.

As we reported ... the family paid respects to Beth Saturday with a public memorial service at Fort DeRussy Beach Park, attended by Dog and his family. During that event, a group also boarded an outrigger vessel and paddled out without Dog on board, and we've now learned why.

Our sources say during the paddle-out -- when everyone was distracted by the final ride -- Dog and his kids quietly walked away, boarded a private boat and headed out to sea to spread the ashes. This was one of Beth's final wishes.

We're told the family didn't spread all the ashes -- they saved some ashes, possibly for another service Dog is planning for Beth's native home state of Colorado.

TMZ broke the story ... Beth died last week at a Honolulu hospital, days after she had been placed in a medically-induced coma. She had been battling cancer for a long time.

She was 51.