They also fear a “perfect storm” of pedestrians injured by uninsured riders finding it more difficult to make claims than if they had been hit by a car.
Unlike in several English cities, there are no trial e-scooter rental schemes in Scotland, although Glasgow is among several councils considering them.
The Forum of Insurance Lawyers Scotland said many riders not wearing helmets were adding to the risk.
Forum member Vikki Melville, who is also managing partner for Scotland at Clyde & Co, said: “The sight of e-scooters is likely to become ubiquitous.
“No restrictions on purchase, despite the current illegality of their use on public roads, is fuelling increased use.
“Whilst Police Scotland has issued numerous warnings, the messaging is not cutting through.
“Despite Scotland having no trial hire schemes currently planned, concerns elsewhere in the UK about increased accidents are just as valid here.
“Approaching summer, expect to see more accidents involving e-scooters, causing often serious injury to riders and pedestrians.
“Riders are often seen without helmets, suggesting complacency about the damage an e-scooter can cause to themselves or others.”
Ms Melville said the forum, which represents Scotland’s 14 main insurance law firms, also had “significant concern” at the lack of clarity over e-scooter insurance cover.
She said while operators were responsible for insurance in hire schemes, illegal users without registration plates or insurance had the potential “to create the perfect storm”.
Ms Melville said: “Perversely, those injured by illegally-operated e-scooters may find a claim to be more complicated than if they had been hit by a larger vehicle.”
“Clarity is urgently needed from government and regulatory bodies as to how e-scooters are to be integrated into existing transport regulations and insurance pricing structures.”
Electric scooters are still illegal on Scotland’s roads – but what’s it like to …
In one case, a marathon runner left unable to exercise after being hit by a teenager e-scooter rider is awaiting compensation six months later.
The 42-year-old Liverpool man said: “I’m 6ft 5in and 16 stone but was hit from behind and wiped out.
“I suffered soft tissue, ligament and tendon damage in my knee, hip and groin.
“I usually run 50 miles a week but have done nothing since.”
Conor Kenny, a solicitor at Thompsons Solicitors Scotland, said: “The current position with e-scooters is unhelpful for everyone.
“They are legal for purchase, but illegal for use in public areas.”
But Simon Hammond, a partner at law firm Digby Brown, said: “While e-scooters are relatively new, a pedestrian injury claim is challenging, but not impossible.
“Depending on provisions within policies, it could be possible to claim against an e-scooter rider’s home or vehicle insurance, but if they riding illegally then an insurance provider may not indemnify the rider.”
Neil Greig, policy and research director of motoring group IAM RoadSmart, said: “The lawyers are absolutely right.
“By delaying yet again the results of the [English] pilot schemes, we have yet another summer looming where people will be buying and using a totally unregulated form of transport in Scotland.
“The pilots were launched in July 2020 and are now not due to finish until March 2022, plus the time required after that for analysis and legislation.
“This has taken far too long.
“Police Scotland should make it absolutely clear that anyone caught riding an e-scooter outside private land will have their vehicle seized immediately.”
Police Scotland deputy head of road policing superintendent Simon Bradshaw said in March: “There will be certain places where this is an issue, but it is not a big problem – at the moment.”