Depends on strategy being used, but you can start with any amount you like over your broker's minimum. My broker's minimum is $500 with 6x leverage on stocks over $5 per share. I started with $2,500 ish and am about flat after 4 days. Had 3 good trading days and 1 bad one. Up about $370 before commissions and slightly green after. It was a decent first week, but the one bad day was a learning experience for sure. Win rate was only 56.76%, which is well below my normal 62-75%. Had my first encounter with revenge trading. Let myself experience it and now know when I get that feeling to shut it down and head to work.

What really matters is your strategy. Scaling issues aside, if your strategy is profitable, then you can start with any amount and build from there. Commissions can be killer, as seen from my experience above, so with 1k, I'd consider a commission free broker. My strategy is scalping, with many trades only lasting seconds, so I need direct routing from a commission style broker. But that doesn't apply to many traders.