A variant method of laminating is single-sided lamination, from Foliant, Banner American and others. These work as do regular laminators, with the film and the card etc. being fed separately into the machine; but the card is pressed into the film and anchored there with some adhesive material instead of being surrounded on both sides by the film. One common use is for the dust covers for books.
There are also printer/laminators, from HP and others, which will produce 1-sided laminated cards: the required data is inked/inscribed on a special data film, which is then hot-pressed onto the lamination film. Some of these devices will also do magnetic stripe encoding. Naturally the manufacturers have all the software you need to run these operations!
The difference between the two systems is that the laminator/printer is restricted to whatever it's programmed to print, whereas the ordinary one will work on whatever it's fed with.
As it's very difficult to peel the data film from the laminator film, such cards readily display attempts to tamper with them. Such machines are best for high-volume applications - for instance, university student cards - with medium-strict security requirements, and are expensive.
Now it is possible to do one-sided lamination on an ordinary two-sided laminator. Obviously some material has to be used in place of one of the films: Kraft Paper can be used; it will stick to the film but can be trimmed off and discarded. Or, two rolls of single-sided lamination film can be used, with two items at once being placed back-to-back to be run through; they must be pretty well exactly the same size. But a problem here is that the laminations will almost certainly curl: single-sided laminators have special design features to take care of this, but using a "Lay Flat" nylon film will help. But even so, there's a gotcha: these films come on a roll core the diameter of which is probably not compatible with ordinary laminators.
Given all this, the only reason to go this way is to avoid the cost of single-sided lamination machines. Perhaps there's a service available locally.