The rules for a wizard are that they can prepare spells that are in their spellbook, and cast ritual spells from their spellbook without preparing them. There is also the cost and time spent to copy a spell into your own spellbook.
But can one wizard add a spell into another wizard's spellbook?
In theory, it fulfils the rule of being in the book-owning wizard's spellbook, but it doesn't fulfil the flavour of the wizard having to decipher and practice the spell for themselves.
To quote the Wizard class (PHB page 115):
[...] Copying that spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation. [...]
Because of this even if another Wizard wrote a spell into your spellbook, it would be written in their notation and thus you could not actually use it. This is because:
[...] Preparing a new list of wizard spells requires time spent studying your spellbook and memorizing the incantations and gestures you must make to cast the spell [...]
You cannot memorize the incantations and gestures of a writing system you can't read, so you cannot prepare a spell written into your spellbook by another Wizard.
Of course, this all only works under some very strict readings of the rules. A GM is certainly allowed to rule otherwise, especially if the narrative of the story makes sense here. If Wizard A knows the notation system Wizard B uses and writes a spell into B's spellbook (which would still have the time and gp costs), I see no reason B couldn't prepare those spells. After all, to an outsider, the spells written by A and the spells written by B are completely indistinguishable.
While there's some wiggle room, this GM (and player) has always interpreted the wizard's spellbook as being their own. 5e does add (strong, I think) support for this interpretation with the Spellbook feature:
Copying [a found wizard spell] into your spellbook involves reproducing the basic form of the spell, then deciphering the _unique system of notation used by the wizard who wrote it). You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.
(emphasis mine)
That is: each wizard notates their spells in a unique way. To cast a spell, the wizard needs to know how to do so - "the sounds or gestures required"; in order to know those things, they must have deciphered the spell and transcribed it into their own spellbook.
If wizard B scribes a spell into wizards A's spellbook, wizard A still hasn't deciphered it, so can't prepare it or cast it.
The sidebar "Your Spellbook" in the description of the wizard class says (PHB, page 114):
Copying that spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.
Not only is the element of "your own notation" important, but the practice is important. It is narratively important, because wizards learn their spells "through dedicated study and memorization".
It is mechanically important because it limits the number of spells a wizard has access to. If the learning portion of spell scribing is removed, at the extreme, a wizard could get others to write down large numbers of spells for them. Again, as an extreme, a wizard could get someone to scribe spells while the wizard is off adventuring, thereby removing one of the limitations on wizards.
As Exempt-Medic pointed out, the GM can choose to rule otherwise, and is fully justified in doing so to fulfill their vision of the game. It's not just flavor, though, and it's not just a very strict reading of the rules. The restrictions on spell-copying for wizards, as well as the various "Savant" subclass features, and the whole Order of Scribes subclass, are all part of the balance of the class. It's probably not game-breaking to allow wizards to copy spells for each other, but it might be game-changing.
The sidebar "Your Spellbook" in the description of the wizard class says:
When you find a wizard spell of 1st level or higher, you can add it to your spellbook [...]
Only you can add spells to your spellbook. So only spells you added can be in there.
First, can another wizard write a spell in a spellbook that is not theirs? Absolutely. A standard spellbook is described in the adventuring gear section of the PHB thusly:
Essential for wizards, a spellbook is a leather-bound tome with 100 blank vellum pages suitable for recording spells.
It is not a magic item: not only does it show up in the list of mundane adventuring gear, it also is not listed in the magic item section of the DMG. It is just a book1 of high quality, that you use expensive, special inks to write spells into. This unofficial Jeremy Crawford tweet confirms it.
So there is nothing stopping another wizard writing spells in "your" spellbook. You just bought the book on a bazaar. It does not mystically get bonded to you by that act. If you find a spell book of a rival mage that has blank pages, you can happily use it to write your own spells into it. (Although the rules do not specify how many pages your spells need).
However, that does not mean you can decipher their spells. As has been quoted in other answers, the sidebar "Your Spellbook" in the description of the wizard class (PHB, page 114) says you can copy spells that you find (e.g. on scrolls or in the spellbooks of other wizards) into your spellbook:
Copying [a] spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.
For each level of the spell, the process takes 2 hours and costs 50 gp. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it. Once you have spent this time and money, you can prepare the spell just like your other spells.
This clearly states
You must transcribe the spell into your own notation (which costs time and money).
Only once you have done so, you can prepare the spell.
This is really the condition for one of "your" spells: it needs to be written down in your personal notation.2 Because of this, if another wizard scribbles his arcane mumbo-jumbo in their notation into your book, they might be able to prepare it from there, but for you they are just wasting your pages. You will be unable to prepare it until you have copied it onto new pages into your own notation. At that point, you can prepare the spell.
1 Although it does not have to be a book.
2 A simulacrum you made of yourself is using your personal notation, and you argue you could prepare the spells it copied into your book. Such a simulacrum, however, is not another wizard; it is an illusory duplicate of the same wizard.
Copying [a] spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.
So I am looking at this other wizards book practicing out of it. And deciphering it, so once I have figured out his notations I now can read the book and it is essential mine. The rule as written is kinda dumb. Because Practicing until i understand it means I now know how to cast it. So if I can cast it I must have been able to read it so, yep I studied it until it was mine.
