On questions related to self-funding a PhD vs. accepting an offer with funding, most answers point out that lacking funding has broader implications for your experience in graduate school - it's not just about the debt. There is a strong consensus that it is not a good idea to accept an offer without funding in any field and country where funding is typical. In some areas, however, such as humanities studies in the United States, a funded Ph.D. is not typical.
In some fields and locations (e.g. computer science in the United States), a standard PhD offer comes with funding. In situations like this, JeffE says:
Do not accept a PhD admission offer without funding. If they really want you, they'll pay for you.
A typical PhD offer from a strong department includes guaranteed funding in some form. My department promises five years of funding to every incoming PhD student, assuming they make steady progress toward their degree. (Do not accept a PhD admission offer without funding. If they really want you, they'll pay for you.) Most of our students take 6 years to finish, but in practice, (100-ε)% of our students are funded for their entire stay. A typical theory student in my department is a TA for 2-4 semesters and an RA of fellow for the rest.
When a student is admitted, the department is making a contractual commitment to funding that student, assuming they make adequate progress toward their degree.
This is echoed in an answer by ff524 on another question.
Choosing the no-support program is a large gamble. You should take it if you're confident that the program made a mistake in not offering you funding -- you have tremendous talent that for some reason has not been revealed in your record. Absent such a situation you should take the support.
On a similar question, L Platts points out that a program offering funding is likely to be more committed to your success, which has broader implications for your experience in the program (not just your debt):
If option A is research council funded (or is funded by a high-profile UK body or another funder demanding results for their money), this would weigh heavily in my decision to take it, even if it is at a less prestigious university. There will be consequences for the group and department if a council-funded student fails to finish by the four-year deadline, and this means that both the supervisor and institution are absolutely committed to the student succeeding and solutions will have to be found if things start to go wrong.
Paul points out that many programs won't even allow self-funded students, for good reason:
In science-related graduate schools, it is quite often the case that students will not be accepted into the program unless they have some sort of support (i.e. department assistantship, scholarship/fellowship, etc...). Students who try to do it all on their own often find themselves under even more pressure than a funded student. On top of trying to pass extremely difficult courses and pursue original, cutting edge research, they may find themselves also working multiple unrelated jobs that barely make ends meet for rent, much less tuition and all other debts incurred along the way. Often, unfunded students succumbs to financial pressures and drop out to pursue more financially stable opportunities.
I believe this is a major reason why self-funded students are often not even allowed in graduate programs: statistically speaking, their success rate is likely too low to merit taking a chance.
ending with this bit of advice:
My advice to you: If you're offered funding, take it!. If you are accepted into a graduate program and are not offered funding and don't have any other source of funding apart from yourself, then don't try to do it all on your own. The sheer cost of graduate school, combined with the uncertainty of you graduating from the program, along with the nightmare of trying to pay off student loan debt for the rest of your life (even bankruptcy will not save you from student loan debt); it's just not worth it to you.
You may be wondering whether it's a good idea to rely on the possibility of getting funding after you enroll in the program. ff524 says that it's a gamble:
It's not impossible to get funding after beginning the program (e.g., if you really hit it off with a potential PhD advisor who has grant money to spare). But this depends very much on luck and circumstance, not just on merit; so unless you like living dangerously, it's not an advisable strategy.
Having said all that, if you do go the self-funded route, Paul has some reassurance for you:
You need not worry about the existence of a caste-system among graduate students. You will not be treated any differently than any other student if you are accepted into a program and not funded.
and Suresh says not to worry about how future hiring committees will perceive this:
All a recruiting committee should (and does) care about post-PhD is the quality of your work (for faculty positions there are additional issues). No one cares about how you were paid to do that work.