Historically, I understand that it was the privilege of any knight to make any of his followers into a knight. (Or anyone at all, I suppose, though obviously it would be restricted in practice.) Is this still the case? In theory, can Sir Andrew Davis or Sir Terry Pratchett or some such modern knight just make anyone they like into a knight?
Alternatively I'd question the premise. Did common knights (as opposed to Barons, Earls, or other titled nobility) ever have the right to bestow knighthood upon others - particularly with regard to England, Great Britain, or the UK? If not, where did this idea arise?
Today all knighthoods are bestowed by the Queen or by a member of the Royal family on her behalf. I have no idea with regard to the past.
As an Englishman I can only really answer this in regards to the Kingdom of England and the subsequent United Kingdom effectively, as my knowledge of foreign knightly orders isn't great.
What I can tell you is that a knight has never been able to make anyone a knight. The only person who can bestow knighthoods and other titular dignities such as peerages is a Fons Honorum (Fount of Honour). This is almost always a monarch, but some individual orders exist with their own Fount of Honour, though these are not considered legitimate by most.
It's also worth bearing in mind that knights always belong to orders and orders have set numbers of members. For example, The Most Noble Order of the Garter has twenty four members and never any more. Other knightly orders, like The Most Honourable Order of the Bath have ranks such as Knight Grand Cross and Dame Grand Cross, Knight Commander and Dame Commander and Knight Companion. Whilst Knights and Dames Grand Cross have influence on the monarch and might informally advise the Fons Honorum on who to appoint, they cannot under any circumstances knight anybody else.
For the first century and a half or so in which knighthood was in existence - perhaps 900 to 1050 CE - it was not a title but an occupation. You were a knight (miles is the period word) if you possessed a horse, mail, a sword, and a lance, and had the training necessary to use them. You could be either a landowner, through vassalage or outright allodial holding, or a dependent member of a noble household.
Gradually, the concepts of knighthood and nobility became enmeshed, and by 1100 nobles were increasingly self-identifying as knights. From this point onward knighthood as a class becomes more formalized, more rigid, and more static. Formal dubbing ceremonies develop, and become more elaborate over time. The concept of heraldry is invented and then expanded upon. By the end of the 13th century knighthood has so many duties and expenses attached to it that it is only available, basically, to very wealthy landowners; the everyday business of war is carried on by men-at-arms, who are knights in all but title. People often confuse these men as being lower class soldiers, but they generally came from knightly families, but could not afford to be themselves knighted. They are frequently referred to as esquires, but as people tend to confuse this with boy servants, I avoid using it in a general discussion such as this. Any esquire's hope was to enrich himself enough to be knighted, but this was increasingly unlikely.