Their doc comments give a hint towards their original intent, written Go's early days:

// A ReadError reports an error encountered while reading input.
type ReadError struct { etc }

// A WriteError reports an error encountered while writing output.
type WriteError struct { etc }

The WriteError type is simply unused by the package:

$ grep WriteError src/compress/flate/*.go
src/compress/flate/inflate.go:// A WriteError reports an error encountered while writing output.
src/compress/flate/inflate.go:type WriteError struct {
src/compress/flate/inflate.go:func (e *WriteError) Error() string {

The ReadError type is sometimes used by the package, but it's not returned consistently enough for users of that package to rely on it. For example, as reported on issue #11856, this program (https://play.golang.org/p/DrL0flbZFg):

package main

import "bytes"
import "io/ioutil"
import "fmt"
import "compress/flate"

func printReadError(lvl int) {
b := bytes.NewBuffer(nil)
wr, _ := flate.NewWriter(b, lvl)
wr.Write([]byte("the quick brown fox jumped over the lazy dog"))
wr.Close()
bb := b.Bytes()
rd := flate.NewReader(bytes.NewReader(bb[:len(bb)/2]))
_, err := ioutil.ReadAll(rd)
fmt.Println(err)
}

func main() {
printReadError(flate.NoCompression) // flate.ReadError used
printReadError(flate.DefaultCompression) // flate.ReadError not used
}

Prints:
flate: read error at offset 27: unexpected EOF
unexpected EOF

Thus, it is highly unlikely that existing code was depending on Read always or even sometimes returning ReadError. On the other hand, the possibility of returning a ReadError complicates test code like https://go-review.googlesource.com/#/c/14832/3/src/compress/gzip/gunzip_test.go which says:

_, err = io.Copy(ioutil.Discard, r)
if ferr, ok := err.(*flate.ReadError); ok {
err = ferr.Err
}

The proposal is to change the doc comments (but not the exported fields and methods) in inflate.go to be:

// ReadError is deprecated.
type ReadError struct { etc }

// WriteError is deprecated.
type WriteError struct { etc }

and to remove any other ReadError references in that package, so that the two occurences of:
f.err = &ReadError{f.roffset, err}
would become:
f.err = err

There are no references to flate.ReadError or flate.WriteError in the standard library or under golang.org/x, other than golang.org/x/tools/imports/zstdlib.go which is an autogenerated file.