More than one in 10 people may feel after-effects, including headache, tiredness and tenderness where the injection was given, reports BBC.
But people must not be deterred from having the vaccine, which saves lives, the Royal College of GPs says.
After-effects are usually mild and disappear within days, experts say.
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But they can still feel quite grim.
David Kidd, 49, from Yorkshire, was unprepared for the cluster of symptoms he experienced shortly after his first dose.
"I had heard that there were some possible ones," he says.
"They do say you might get flu-like symptoms but I was not expecting to get lots of them together.
"I had the vaccine at 12:00 on Saturday and I felt fine for the rest of the day, until about 18:00 that evening."
Mr Kidd had:
body aches
a fever
a headache
sore eyes
nausea
a temperature
freezing cold extremities
"It felt like I had plunged my hands and feet into ice or a freezer," he says.
"It was really painful and my head was banging away. It was the worst headache I have ever had in my life.
"And my eyes were burning, along with the obligatory nausea.
"I couldn't get comfortable all through the night."
'Better prepared'
Mr Kidd took paracetamol and ibuprofen and, by Sunday evening, began to feel a bit better.
And it has not deterred him from going for his next, booster shot.
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"Luckily, I was ill over the weekend," he says. "So it did not cause too much bother.
"But I will be better prepared next time, just in case for my work.
"And I will make sure I have got some shopping in."
Feeling sick
Side-effects that may affect more than one in 10 people after the Oxford-AstraZeneca vaccine or the Pfizer-BioNTech shot include:
tenderness, pain, warmth, itching or bruising where the injection was given
generally feeling unwell
feeling tired (fatigue)
chills or feeling feverish
headache
feeling sick (nausea)
joint pain or muscle ache
Side-effects affecting up to one in 10 include:
swelling, redness or a lump at the injection site
fever
being sick (vomiting) or diarrhoea
flu-like symptoms, such as high temperature, sore throat, runny nose, cough and chills
Allergy to the vaccine is very rare.
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Medical help
Prof Martin Marshall, from the Royal College of GPs, said: "Patients should be assured that both Covid-19 vaccines currently being used in the UK have gone through a rigorous process to ensure they are both safe and effective.
"It is normal, as with most vaccinations, for some patients to experience mild side-effects.
"If required, we would recommend that patients treat these with painkillers, such as paracetamol."
However, if symptoms become much worse or cause concern patients should seek medical help by calling 111, their GP practice or, in an emergency, 999, he added.
The Medicines and Healthcare products Regulatory Agency's Yellow Card scheme allows people, such as Mr Kidd, to report any vaccine side-effects.
And those logged for the millions of Covid-19 shots given were "broadly similar" to other vaccines in routine use, chief executive Dr June Raine said.
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'Greater protection'
Meanwhile, data from the Zoe Covid Symptom Study - which has received feedback, submitted via an app, from more than 280,000 people - suggests fatigue, headache and chills are more likely to be experienced by: women, younger people, those who have had Covid
Younger people and the slightly more women than men who have more side-effects may just have fitter immune systems, lead scientist Prof Tim Spector, at King's College London, said.
And this strong immune reaction may provide even greater protection from just a single shot of the vaccine - although people should still have the recommended two doses.
"If someone has had previous Covid, it is a bit like they are primed," he said.
"And their body is ready to fight it again."
The Yellow Card data, however, shows no link between Covid infections and vaccine after-effects.
But it aims to record any severe reactions rather than every mild one.
And Public Health England says people who have had Covid should still be vaccinated, to protect themselves and others.
Prof Spector said: "You are going to be less ill than getting Covid.
"Do not delay or avoid getting your dose.
"We are urging as many people as possible to download the Zoe app and log their vaccines with us, so that we can independently monitor how we all react differently and how the vaccines impact the overall pandemic."