You need to talk to a lawyer.
Laws surrounding the ability to enforce a non-compete agreement vary by state and country. Where I'm at the state supreme court has ruled them as essentially unenforceable unless there is corresponding "consideration" for signing. Appropriate consideration here does not include the actual job itself or even being privy to confidential information. From what I've seen about the only thing that passes muster is if the company is offering you some form of compensation above and beyond the normal pay and that it is tagged as such.
Regarding "triggering" a non-compete I'm not exactly sure what you mean by that. You either have an agreement or you don't. If you do then it is obviously in force the moment you leave, the company doesn't have to do anything. They just let the lawyers sue you and the target company when they find out where you went. Believe me, they will.
I will say that it if this agreement is enforceable at your location then you should inform the new company about it prior to accepting their offer. The likelihood of the new company continuing to hire you would then be remote. However if you don't inform them then you are opening yourself to a whole can of lawsuits and other troubles from everyone.
Regardless, you need to talk to an attorney as soon as possible.
From the comments, it sounds like it is likely enforceable - to be clear I'm not a lawyer so you should seek appropriate counsel.
Assuming it is then I would first contact this competitor and verify they are hiring. During that process I would tell them about the non-compete and let them know my plans to have the current company throw it away. See where that takes you. If you have a job offer - likely contingent on the outcome of the non-compete, then approach HR / your manager at your current company and state your intentions.
You must get a release from the non-compete in writing - verbal assurances are NOT enough no matter who else is in the room. Assuming they do this, then you're good. If they don't then you need to inform the new company that you were unable to get a release - which will most likely cause them to terminate the offer. Even if they don't then you have to decline. Either way you will have burnt your bridge with the current company so expect to have a bit of R&R.