I can recount three stories about this—so you can judge for yourself.
At my high school in Jerusalem in the 1970s, there were two English teachers: one was from Russia, the other one hailed from Austro-Hungary (gives you an idea how old she was).
As a result, half the school learned English with a Russian accent, and the other half with a strong Viennese accent. Neither half could understand each other in English.
One day, the Austro-Hungarian teacher invited me, as a native English speaker, to come in and recite a passage from Shakespeare before the class.
When I finished, there was complete silence in the room. Finally, someone raised their hand at the back.
“Teacher—what language was that?”
My accent, as a native English speaker, had thrown them completely, and they couldn’t understand a word I had said.
A few years later, as a student in London, I started dating a young Italian woman who was majoring in Arabic at the University of London’s School of Oriental and African Studies (SOAS), Her minor subject was Hebrew, so I offered to help her with her homework.
Looking at her work and at the corrections that her teacher had made, I discovered that her Hebrew teacher was not an Israeli, but a young Jewish English woman who had spent a couple of years there, and had a highly imperfect command of Hebrew—so bad so that she made mistakes in the corrections she made to my friends’ homework. (Shame on you, SOAS! I hope their teacher selection process has improved since then…). Amusingly, my friend and I had only just started going out, and she didn’t know that I was Israeli (judging from from my accent, she thought I was just an ordinary English Jewish guy), so she didn’t know whether to believe me or her teacher…
Two years later, I had a spare room in my flat, and I decided to rent it out at a discount to a native French speaker, on condition that they speak French to me (to improve my French). A young woman answered my advert, told me she was Belgian and her native language was French. I took her at her word, but a couple of weeks into the tenancy I caught her making errors that even I knew were wrong, so I confronted her, and she admitted that her native language was Flemish. I let her go, and a young woman from Lyon took her place, which made all the difference.
Bottom line: go with someone native—or as near as dammit.
(Originally written in reply to a question at Quora.com).