Magic weapons, like all cool magic, should be rare and special. The average person won't even have a sword or other cool weapon (remember, only 1% of humans are adventurers). But what about people that would have weapons as part of their jobs, such as soldiers and policemen?
These fellows are technically fighters by class, but I would not consider them part of the 1%-ers. I consider them to be experts in human-to-human combat, equipped with standard-issue, minimal, for-urban-use, gear. Over the years, they could rise in level, like the MM1e describes larger bands "led by the 3rd level fighter". But, to me, that is just their job.
It's like getting your airplane mechanic's license. Guys can be jet repairman in the Air Force, but that does not help them get their A&P license. Those military guys are considered specialists, while to get your A&P license, you have to be a generalist.
I think of the Fighter class like that. You are not just a guy with skill with a sword, you are not just a "town watch" guy, you are not just a guy with a spear marching in an army. You are an ADVENTURER! You have training in multiple weapons, you have to learn general survival skills, you have to fight terrifying MONSTERS, not just other people, etc.
Only an adventurer would ever have a magic weapon. A town policeman, or palace guard or whatever, would never have one, even if he was the proverbial "7th-level captain of the guard". A great asset in a brawl or scrum he would be, no doubt, but not made of the same mettle as an adventurer. He would never have any occasion to need, or be awarded, a magical weapon. He is purely mundane, non-heroic.
Weapons from Wizards
An adventurer would come into a basic magic weapon most likely as payment from a high level magic user. As I discussed in my post about the magical economy, wizards are always in need of magical ingredients. Magic items take a long time to prepare, so they can't spend all their time outside, nor would they want to face the danger. Since they can command so much money for their finished magical products, they can pay good money for the necessary ingredients, and hire the adventurers to do the dirty work.
Rather than cash, the needy wizard might offer a magic item in payment. Or conversely, the magic user might simply give the magic item to the PC if the PC would absolutely require it to get the job done.
Weapons from Holy Orders
A magic weapon could also come as a reward for assistance to a clerical organization. Clerics would naturally produce these magical weapons as part of their never-ending war against demons, devils, and undead, most of whom cannot be harmed by non-magical weapons. Low level clerics could be given one as part of their initiation ceremony into 2nd or 3rd level or something.
Fighters or other classes might be awarded a magic item by the clerics if they take some oath or pledge to help fight evil, or something, or help the clerics on some particular quest against spiritual darkness.
+1 type weapons would be considered standard and available to low level adventurers. But what about the higher pluses? The DMG gives us a hint how this would work, in its description of +3, +4, and +5 armor, which it says is constructed out of different, rarer, even "otherworldly" material (meteoric iron, adamantium, etc).
More Powerful Magic Weapons
I am thinking that multiple pluses on any magical item have to be earned with some special task or quest.
For example, let's say you somehow come into possession of the magical formula for charging your weapon with "supernatural power". Maybe you find it in a book, maybe an old wizard gives it to you, whatever. The task: to wrap your weapon 7 times around, in the fresh hide of a juvenile dragon, then leave it upon a consecrated alter upon the tallest mountain, under the light of the full moon. Whammo, your +1 sword is now +2!
I am thinking a +3 weapon, a la the DMG suggestion, would require special construction from the start, made out of some special material (like can only be purchased in person from the ancient dwarven smiths deep under the mountains). Then it would have to be enchanted like normal to +1, THEN "charged up" according to some special magical formula, perhaps twice.
Gifts from the Gods
I would postulate that +4 and +5 weapons are solely the domain of the lesser and greater gods, respectively. Only they would be able to forge them from such otherworldly materials, then imbue them with such celestial power, to give them their permanently devastating powers. Of course, the gods would only grant such weapons as part of some greater plan, in return for the completion of some quest or heroic deed.
Special-purpose weapons, the ones with special bonuses against particular monsters (like dragons, giants, lycanthropes, etc) could be created from a fairly standard formula: press the weapon for six days in the oozing skins of the monsters, then boil the weapon for 6 days in the blood of the monsters, then, spread the powered dust of the monsters' bones over the weapon, continually for six days.... That would give them the proper taste for the flesh, blood, and bones of that particular monster, I reckon.
Weapons with Personalities
Weapons with personalities of their own? To my mind, those involve SACRIFICE. Using some profound, and possibly gruesome, ritual, the soul of a living being is transferred into the magical weapon. The person was so fanatical about their mission in life, they willingly entered into a weapon, in order to be able to continue to advance that mission, forever! Naturally, they will attempt to domineer anyone holding them into a complete devotion to that task as well...
Or perhaps a greater god created a soul from scratch and put it directly in the weapon. As an infant, the weapon is of course out to prove itself and see the world while fulfilling its divine commission with great zeal, causing who knows what complications to the player character who attempts to weild it to his own purpose!