Loosehead Barrington, 26, has featured 82 times since joining Sarries from Jersey in 2013, and has agreed a three-year contract at Allianz Park.
USA international Lamositele, 21, has signed a new two-year deal.
"Both Richard and Titi are valuable members of our squad," director of rugby Mark McCall said.

The new banknote will be smaller than the current one and be printed on a thin plastic.
For more than 100 years, English banknotes have been made from cotton paper.
Polymer banknotes are made from a very thin, flexible, see-through plastic film, with the design printed on special layers of ink on the front and back.
Because the main material is see-through, the design can include clear spots that are like little windows you can look straight through.
More than 20 countries around the world already use polymer banknotes - they include Scotland, Australia, New Zealand and Canada.
The Bank of England says that polymer banknotes will be better than paper, because they will be:
Cleaner: Shiny plastic doesn't get wet and dirty like paper.
More secure: Features like the see-through windows are difficult to copy, so it's harder for criminals to make fake notes.
Longer lasting: Plastic won't tear as easily as paper, so the notes should last longer.
They're even expected to survive if they accidentally go in the washing machine, although they'd melt under extreme heat.
The new £5 note will be going into circulation in September this year.
And it's not just the £5 that's going plastic - polymer £10 notes are planned for 2017, and polymer £20 notes should be available by 2020.

Abdul Nabi, 30, is described as black - of Asian appearance - about 5ft 7in tall and skinny, with short curly hair.
Anyone who spots him has been asked to call 999.
Officers were searching land around Midpark Hospital and Maidenbower near the Crichton university and business complex. Premises in the immediate area were alerted.
The search subsequently moved on to the Calside and Georgetown areas of the town and pupils at Georgetown Primary School had their sports day interrupted.
Police dogs and the police air support unit also joined efforts to try to trace Mr Nabi.
"This is a missing person inquiry and we want the public to be alert, however not alarmed," said a spokesman.
"We are asking that if he is seen he is not approached and you contact the police."
Insp Rory Caldow said officers had been carrying out a large-scale search since about 12:30.
"We did have early concerns that he may have been in possession of a knife, however we understand that this is not now the case," he said.
"Mr Nabi is a vulnerable missing man and our concern for the safety of the public centres on the uncertainty of his reaction should he be approached.
"We ask the public to be vigilant and report any sightings, without making an approach to him, on the 999 emergency number."

15 June 2013 Last updated at 17:21 BST
Tafwyl is a free family festival in the grounds of Cardiff Castle, featuring music, sport and craft activities, as well as cookery and literature. Around 10,000 people were attending.
Rhys, who is also due to start filming a BBC Wales drama from Monday, said holding the festival in the city helped to break down potential barriers to the Welsh language.
Education Minister Leighton Andrews also said it had an important role to play in showing off the language and its growth in the city.
Prop forwards Richard Barrington and Titi Lamositele have signed new deals with Premiership and European Champions Cup winners Saracens.
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The Bank of England has unveiled the design for the new plastic £5 note.
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Police have warned the public not to approach a man reported missing from a psychiatric hospital in Dumfries.
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Actor Matthew Rhys, the star of Cold War drama series The Americans, returned to his home city Cardiff to open a Welsh language festival.