Playing flute right handed or left handed?

Yes, it is Ok for a right handed person to play flute left-handed. I would not use playing left-handed or right-handed, but will use the word grip.

For example, the world famous flutist from India, Pandit Hari Prasad Chaurasia had a left-handed grip. He changed his grip to right-handed as a gesture of respect to his guru Annapurna Devi.

From Wiki:

Much later, while working for All India Radio, he received guidance from the reclusive Annapurna Devi, daughter of Baba Allaudin Khan. She only agreed to teach him if he was willing to unlearn all that he had learnt until then.

Another version is that she only agreed to teach him after he (of his own) took the decision to switch from right-handed to left-handed playing to show her his commitment.

In any case Chaurasia plays left-handed to this day.

Full story on his interview at FirstPost

I am left-handed and when I learnt flute, I picked and held it exactly like Pt. Chaurasia in below picture - flute to my left - whereas my dad and brother play holding it like Lord Krishna.

Image source: Wikimedia commons

Right-handed flutists :

Lord Krishna, a Hindu God:

Image source: Flickr

Image source: Wikimedia Commons

Footnotes

[1] Learning from the master: Corporate lessons from flute maestro Pandit Hariprasad Chaurasia - Firstpost

[2] Image on wikimedia.org

[3] Image on staticflickr.com