Bedfordshire Police recorded 24 incidents of sexual communication with a child between April 2017 and September 2018, according to Freedom of Information requests from the NSPCC.
Facebook, Snapchat and Instagram were used in half the cases where police recorded the method of contact, with seven victims over the 18 months.
Girls aged between 12 and 15 were the most likely targets, but one in four victims were 11 or younger, according to the reports which also include an age.
The NSPCC has called for tougher measures to tackle grooming, after its investigation found more than 5,000 reports of sexual communication with children across 39 police forces.
The offence, which carries a maximum sentence of two years in prison, was introduced in 2017 to try and capture online-only sexual offenders.
NSPCC chief executive Peter Wanless accused social media firms of "10 years of failed self-regulation".
He said: "These figures are overwhelming evidence that keeping children safe cannot be left to social networks.
"We cannot wait for the next tragedy before tech companies are made to act."
Mr Wanless said that a sharp spike in grooming offences on Instagram was "hugely concerning", urging the platform to add more protection for young users.
The use of Instagram to groom children in England and Wales has tripled - there were 126 cases between April and September 2017, rising to 428 over the same period last year.
Between April and September 2018, at least one child in Bedfordshire was targeted through Instagram.
A spokesperson for Facebook and Instagram said: "Keeping young people safe on our platforms is our top priority, and child exploitation of any kind is not allowed.
"We use advanced technology and work closely with the police and Child Exploitation Online Protection Command to aggressively fight this type of content and protect young people."
The National Crime Agency said it was crucial that online platforms also make it easy for children to alert them to potential offending and report sexual abuse.
A spokesperson said: "The NCA helps industry to enhance their reporting tools and, where possible, shares knowledge and expertise to support the industry to improve standards and security online."