When should we as parents teach our child to use scissors for cutting a piece of paper? At what age typically can a child use scissors properly, i.e. with one hand?
You can get blades woth sprung handles and short blades. Here's one example. http://www.amazon.co.uk/gp/aw/d/B00366ICX2/ref=pd_aw_sims_2/276-6568127-1578546?pi=SL500_SS115
Young children will not know what handedness they are and it does make a difference. Young children are likely to be frustrated by this.
Also, many adults also use scissors incorrectly (some scissors need you to keep your index finger out of the loop) and pass bad cutting habits on, which again frustrates children.
The correct grip requires the hand to be as if the child was about to shake hands, and not palm-down which they usually have. It also requires stretched index finger and thumb. These are tricky for children to achieve.
Also, you don't mention supervision. When supervised cutting paper is fun and useful. It helps develop fine motor skills.
So, teach correct grip; encourage practice even when child is failing (because effort should be rewarded); manage expectations so child is not disheartened with poor results; and supervise young children.
Here's a nice link: http://www.schoolsparks.com/blog/teaching-children-the-proper-scissors-grip
So I would say when the child can hold scissors correctly, and manipulate the scissors and the paper. This is likely to be around the age of about three, but will require close supervision and special scissors. Younger children don't have the finger strength nor dexterity.
You can, and should, get fully plastic scissors that are capable of cutting paper but are completely safe (as far as any toy is), at around 2. My almost-three year old has been using them for six months or so and not only loves using them but has become very adept at it (he can cut with one hand properly most of the time). This was one of the milestones for the daycare we used to go to during the 2-3 year old range.
One example of these is Crayola's My First Safety Scissors which are officially recommended for 3 years + ("choking hazard"), but as with most things is fine at 2 in my experience as long as it is used under supervision (and I can't really imagine what could fall off of these to choke on).
There are scissors which will cut paper, but have some protective element for reducing the likelihood of harming a person. Training children in their use begins around 4-5yo with supervision. As their skills improve, other more serious scissors may be used.
I personally keep myself away from those super-slick ones that will cut through anything... I make mistakes too easily.
Start with safer and progress to more-care-needed as your evaluation of the child's skills warrants a change. There is no substitute for judgement based upon personal review.