In my personal opinion (John Loughran) children should only be allowed on a device (phone, tablet, laptop etc.) that can access the internet, under parental supervision, in a family room in a position where a parent can see the screen and can check with a glance that the child has not gone to an unapproved website, or is communicating with strangers. In addition screen time should be limited to what you feel is appropriate for your child. In general I try to limit my kids to about 40 mins per day, though I do make exceptions at times if it is educational and interactive e.g. chesskid, typingtutor, minecraft. I also try to balance it with physical and outdoor activity and exercise as well as indoor jobs and creative play. Kids also need to be taught good practice like not clicking on ads, not communicating with strangers, not opening the Smith's catalogue! :-) Of course other parents may not share my opinions on these matters.
While chesskid.com is pretty child-safe, other chess websites do allow limited chatting facilities with strangers and may need monitoring, depending on your approach as a parent, and the age and maturity of your child.