I am sorry if this is a duplicate, but formatting whit JSON became lot complicated and I recognized that:

• Tellraw (doesn't accept \ ):

  /tellraw @p {text:"HI"}
  

• Signs (needs \ ):

  /give @p sign 1 0 {BlockEntityTag:{Text1:"{\"text\":\"\",\"extra\":[{\"text\":\"CLICK\"}]}",id:"Sign"}}
  

• Mixed (doesn't work, which makes pretty much sense):

  /give @p sign 1 0 {BlockEntityTag:{Text1:"{\"text\":\"\",\"extra\":[{\"text\":\"CLICK\",\"clickEvent\":{\"action\":\"run_command\",\"value\":\"/tellraw @p {text:\\\"Hi\\\"}\"}}]}",id:"Sign"}}
  

  (I made this using: http://ezekielelin.com/tellraw/#addExtraModalData )

This will cause the game having problems with text detection which will cause the game to put the whole command on the sign. Without the \ it wont recognize the text in the /tellraw as text and will give (in a command block) as ouput:

[13:28:53] Kann NBT-Daten Unexpected token 'H' at: Hi"}\"}}]}",id:"Sign" nicht analysieren