As you are suffering the results of the flooding the burden of sorting this out will probably fall to you.
The blockage may not be your fault but due to one or more of the other residents. I don't think that you should take on dealing with this yourself on a DIY basis. Can you find out if everyone is aware that this drain is only a soak-away & that it cannot be abused? Do you have new neighbours that could unwittingly have caused the problem? Or has the drain just reached it's capacity & too much water is now being put down it? Call a meeting of your neighbours & explain the problem. Having done your research including costing for different options & possible legal implications, propose some solutions. Give every household written information on each option to take away with them.
Get a drain expert in. Find out what the problem is. Discuss solutions & costs. Vote & then act. Does the drain need digging out & replacing? Or should everyone now have individual drains installed? Should these be soakaways or septic tanks? Etc.,
Hope this helps.