The 17-year-old boys, who were discovered at Huntcliff in Saltburn at around 7pm on Friday, are Harry Watson and Alex Yeoman.
Police said inquiries are ongoing to establish what happened to the boys, who are both from Loftus, and that the incident is not being treated as suspicious at this time.
Harry and Alex are believed to have attended Freebrough Academy, a specialist engineering college in Brotton.
The mother of one of two teenagers found dead at the bottom of cliffs has said he was "one of the gifts of life".
Harry's mother, Tanya, posted on Facebook: "It is with great sadness and sorrow that last night our precious 17-year-old son Harry Watson has passed away."
She said: "Please understand at this difficult time we don't have many answers but Harry is one of the gifts in life always there for everyone and anyone.
"He will always live in our hearts and we will get through this as the strong family that we are because that's what Harry would have wanted."
In a statement through the police, Harry's family said he was "precious" and would be sadly missed.
Alex was described as "a loving son" who would also be sadly missed.
A Cleveland Police spokesman said: 'Specialist officers are continuing to support both families.
'The families have expressed their thanks for the messages of support and condolence that have been sent to them, however, they would like to be left to grieve in peace.'
Floral tributes have been left at the scene for the boys. Police are asking that any other floral tributes are left at the bottom of the steps to Huntcliff.
Any witnesses who may have been in the area of the Circle Sculpture at Huntcliff around the time of the incident are asked to contact Detective Sergeant Paul Hodgson from Cleveland Police Major Crime Team on the non-emergency number 101.