Yes, even if you're in a committed, long-term relationship. And so should your partner.
This is inspired by a few threads I've seen recently.
It's not mortifying, or embarrassing, or a moral failure. It does not make you dirty or unclean if you have an STI. I see so many threads on here where someone is horrified at having to go for an STI check and they've never ever had one before. Please don't be. It's just another part of basic health maintenance which we should all be doing like dental check-ups.
If you are or have ever been sexually active, you should be getting tested regularly, from every couple of years or so to every few months. Regardless of sexual orientation, number of partners, or protection used. You (and your partner - don't take no for an answer if you want to move to unprotected sex) should especially get tested before starting a new sexual relationship when you're planning to use contraception other than condoms or femidoms.
You wouldn't neglect any other part of your health, or your partner's health, the same way.
STIs can lie dormant for many years. You can be in a faithful, long-term relationship (an STI is not, in and of itself, proof of infidelity) and still nasty little buggers like genital warts, herpes, or even syphilis can raise their heads suddenly from that unfortunate one night stand back in your first year at uni.
Chlamydia is asymptomatic in ~50% of women and ~70% of men. It would be surprisingly easy for both partners in a relationship to have it and for neither to have the faintest idea. You can carry this for years and it can have seriously detrimental effects on your fertility among other things.
Other infections can have similarly low-level symptoms that you may not even notice.
www.nhs.uk/common-health-questions/sexual-health/how-soon-do-sti-symptoms-appear/
Tests can also fail or provide false negatives, usually due to poor sampling or the test simply not looking for a specific infection.
So for gods sake, please start getting yourselves tested. It should be the rule and not the exception. In lots of places in the UK, you don't even need to go to a clinic (especially pertinent now with covid), you can do a free NHS test at home. These usually consist (for women) of a vaginal swab and a finger prick blood sample. It takes 5-10 minutes. Assorted links below:
National service which works with the NHS to deliver free home test kits: sh24.org.uk/about-sh24#operating-regions (note that this is an NHS-affiliated service and is not comprehensive. I, for example, live in London and can't use this service, however I can use the link below!).
London: www.shl.uk/
Berkshire: www.safesexberkshire.nhs.uk/sexual-health-matters/order-sti-home-test-kit/
Hampshire: www.letstalkaboutit.nhs.uk/worried-about-stis/order-a-test-online/
East Midlands and East Anglia; www.icash.nhs.uk/contraception-sexual-health/postal-self-test-kits
Oxfordshire: www.sexualhealthoxfordshire.nhs.uk/sti/kits/
West Sussex: www.sexualhealthwestsussex.nhs.uk/order-a-free-online-sti-self-test-kit-here/online-testing-information/
There are lots more, so check the NHS trust in your local area. The kits are not always completely comprehensive, some trusts do kits which target only the most common infections, but are a good check to keep up every year.
Of course, if you have symptoms, you must go and see an actual doctor. And make sure you know what you are being tested for - STIs can be missed if the tests aren't tailored correctly. So if you have reason to suspect you may have a specific infection, tell your doctor.
And in a similar vein, please also go for your smear test! (NB: having done several of both, I can say that STI checks are a walk in the fucking park compared to a smear test).
But please don't be embarrassed, or feel like it's something to be judged for. You wouldn't feel awkward over a blood pressure check. It's just another part of your body which needs looking after!