The talks were held at the North's state guesthouse, Paekhwawon, where Moon on Tuesday spent the first night of his three-day trip.
In the first round of talks the previous day, Kim thanked Moon for brokering his summit with U.S. President Donald Trump in June and expressed hope for further progress in the talks with the U.S.
Moon highlighted the great responsibility he and the North Korean leader owed to their nation and the entire world.
"I feel the great weight we must bear, along with a heavy responsibility. I wish this will be a summit that produces abundant results as a gift to the 80 million people of this nation for Chuseok," the South Korean president said, referring to the upcoming traditional holiday that falls on next Monday.
The Moon-Kim summit is the third of its kind. The two leaders earlier met in the border village of Panmunjom on April 27 and May 26. The Pyongyang summit is the fifth inter-Korean summit in history.
Depending on the progress made in their talks, a joint announcement of the outcome may be made, said Yoon Young-chan, the senior secretary to President Moon for public relations.
"It is still difficult to predict how an agreement will be reached between the two leaders," he said in a daily press briefing in Seoul.
"The announcement of the outcome is also scheduled though we do not know the exact time," he added.
Should the leaders decide to hold a joint press conference, it will be broadcast live, the official said. The leaders will likely not take any questions.
A joint announcement of summit outcome may indicate a possible breakthrough in efforts to denuclearize the North.
Ahead of his first-ever trip to Pyongyang, Moon says the No. 1 objective of his visit is to break the deadlock in the denuclearization talks between the U.S. and North Korea.
The talks have stalled since U.S. President Donald Trump voiced his apparent disappointment with North Korea, citing what he called a lack of progress in the North's denuclearization process, also calling off a scheduled North Korea trip by Secretary of State Mike Pompeo in late July.
Moon has urged sincere efforts on both sides to move forward the denuclearization process, stressing the need to find a compromise.
"(My) North Korea trip would have a great meaning if it could lead to the resumption of North Korea-U.S. dialogue," Moon was quoted as telling his aides before heading to Pyongyang on Tuesday.
Officials from Seoul's presidential office Cheong Wa Dae said Moon and Kim may hold a joint press conference in Pyongyang, depending on the outcome of their summit.
Moon will attend a farewell dinner later in the day that could possibly be hosted by Kim.
He is scheduled to return home early Thursday, but Cheong Wa Dae officials said his departure may be delayed for a friendship event with the North Korean leader, again pending on the outcome of their summit in the communist state. By The Yonhap.