Is there any way to broadcast a part of your screen as webcam on OSX (for twitch, omegle etc)?
You can use OBS, also known as Open Broadcaster Software.

Here's the official site's description of OBS:


  Open Broadcaster Software is free and open source software for video recording and live streaming. Supported features include:
  
  
  Encoding using H264 (x264) and AAC.
  Support for Intel Quick Sync Video (QSV) and NVENC.
  Unlimited number of scenes and sources.
  Live RTMP streaming to Twitch, YouTube, DailyMotion, Hitbox and more.
  File output to MP4 or FLV.
  GPU-based game capture for high performance game streaming.
  DirectShow capture device support (webcams, capture cards, etc).
  Windows 8 high speed monitor capture support.
  Bilinear or lanczos3 resampling.
  


You can download OBS for macOS 10.8+ from the official website. It also supports Windows 7/8/10 and Linux.