Is synchronized keyword necessary, in Java, if everything that is modified in a function is local to that function?

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No, it is not required if you make sure whatever parameters are updated/modified will be done as atomic references/operations. Atomicity is an alternative to syncronizatiom.
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In some cases It will be necessary. If you want to develop a thread-safe application then definitely you will have to use synchronized keyword.
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Trainer

Not necessaring, if sharing is not there then synchronization is not required
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If the function only works with local variables, then synchronized is not required. For example, a simple function for addition - "int add(int a, int b)", which takes two integers as parameter and returns summation of them, is not required to be synchronized. But if the function works on some class level...
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If the function only works with local variables, then synchronized is not required. For example, a simple function for addition - "int add(int a, int b)", which takes two integers as parameter and returns summation of them, is not required to be synchronized. But if the function works on some class level properties, global/static variables, or makes some calls to database system, or some I/O operation, then it may be required to make it synchronized to make the program thread safe. read less
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Trainer

No, not necessary
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The synchronized keyword is used in two primary contexts: as a method modifier to mark a method that it can only be executed by one thread at a time. by declaring a code block as a critical section – one that’s only available to a single thread at any given point in time
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The synchronized keyword is used in two primary contexts: as a method modifier to mark a method that it can only be executed by one thread at a time. by declaring a code block as a critical section – one that’s only available to a single thread at any given point in time read less
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