Actually, I think there's interesting discussion to be had around where you're coming from, and where I'm coming from. And, also with regard to the actual law.
Local police departments actually are not supposed to enforce immigration laws. It's even likely unconstitutional in some cases for local police officers to target people for searches and ID checks based on the assumption that they are here illegally. It's not even about my beliefs on the issue. It's about local police policy and ordinances.
The reasons that local police departments aren't tasked with immigration enforcement are many.
For example, imagine a domestic dispute between a person who is in the US without documentation and a partner who is here legally. Let's imagine the partner who is here legally is violent, and beats the crap out of the person who is here without documentation. How do you want this to play out? Do you want the person being abused to not call for help for fear of being deported? Or, do you want the cops to come do their job, help someone who is being hurt, and haul the abuser away leaving the victim of the abuse safe? Most jurisdictions in the US have chosen the latter. Cops have a lot of leeway to be assholes in this circumstance, but they're under policy guidance to help the victim and not make the situation worse.
Another example, say an undocumented immigrant witnesses a violent crime. Do you want them to be deported if they come forward and identify the assailant? Or, do you want them to be able to report the crime, give their testimony (even in court), and help put a dangerous person away?
I believe being part of the civic discourse, up to and including participating in a protest, march, rally, etc. is of a piece with these other examples. If a local police officer catches an armed robber, and that robber turns out to be an undocumented immigrant, I won't argue about immigration getting involved. But, if someone wants to speak to encourage more kind immigration law that effects them and their family? Yeah, I think they should be able to do that, without fear of being targeted.
Civil disobedience is great, and I support it. But, it is also an action that requires privilege. Many undocumented immigrants have families to support (often including children who are legal US citizens, having been born here); if they are arrested and deported for speaking out, their whole family suffers.
Admittedly, I fall very far on the side of being opposed to our immigration laws as they currently stand. I believe state borders are immoral, in and of themselves, regardless of how they are enforced. Even so, the reality is that local law enforcement shouldn't be using the threat of deportation to stifle dissent. Not because I say so, but because people all over the country have said so, and it is reflected in local laws and police department policies. But, when it comes to protests...the rules get tossed out the window. Protesters are nearly always considered second class citizens, and subject to more aggressive tactics; constitutional protections are considered much lower during a protest action, especially if some protesters are breaking the law.
Given that undercover cops often (so often that it is to be expected, and activist trainers have to cover it as though it will happen) incite law breaking in order to allow uniformed officers to engage more aggressively, I think it's time to turn that tide way the fuck back.
In short, I hear you. I might have even agreed with you at some point in the past. I understand where your position comes from. But, somewhere along the way, my allegiance flipped. I simply don't trust my government to have good intentions.