Are they separate counters on a single creature, or do they just cancel each other out? I know,in effect,they just cancel each other out, but are both counters technically still on the creature? Can you proliferate the creature? Does the creature still count as a creative that has that counter on it? Or like with Kitchen Finks, can you put a +1/+1 counter in it to remove the -1/-1 counter?
If a creature has +1/+1 and -1/-1 counters on it and a toughness of 0 or less as state-based actions are checked, it will simultaneously go to the graveyard and have pairs of a +1/+1 and -1/-1 counter removed from it (C.R. 704.5f, 704.5q, 704.3). Even so, however, that creature's undying or persist ability won't trigger, since undying and persist are both leaves-the-battlefield abilities (C.R. 603.6c; review C.R. 702.93a and C.R. 702.79a, respectively) and, therefore, they check the state of the game just before the creature left the battlefield (C.R. 603.10a, 603.10, 603.4; see also C.R. 704.7) and see, as a result, that the creature had a +1/+1 or -1/-1 counter on it, respectively, at the last moment it was on the battlefield.
The +1/+1 counters and -1/-1 counters on a permanent are removed in pairs of a +1/+1 and -1/-1 counter as a state-based action (C.R. 704.5q), which happens any time a player would get priority (C.R. 117.5). The only moment it can happen that you proliferate (C.R. 701.27a) while a permanent has +1/+1 and -1/-1 counters on it is if, before you proliferate—
one or more counters of both kinds are put on a permanent, or
one or more +1/+1 counters are put on a permanent with a -1/-1 counter on it or vice versa,
and both actions happen during the resolution of the same spell or ability (an example is Grim Affliction), because players don't have priority while a spell or ability is resolving (C.R. 117.2e).
EDIT (Nov. 11): One rule was moved and spread out in Commander 2016.
EDIT (Apr. 5): Correction; added rule citations.
EDIT (May 12): Correctness edit.
EDIT (May 16): Edited post to conform to "put" rather than "place" convention introduced in Amonkhet.
EDIT (Sep. 28, 2017): One rule was renumbered with Ixalan.
EDIT (Nov. 29, 2017): Corrections to rule citations.
EDIT (Jan. 21, 2018): Add rule citation.
EDIT (May 3, 2019): Clarification.
EDIT (Jun. 8, 2019): Correctness edit.
EDIT (Dec. 25, 2019): Correctness edit. Some rules were renumbered with Core Set 2020.
EDIT (Oct. 24, 2022): Edited, including because some rules were renumbered in the meantime.
The +1/+1 counters and -1/-1 counters on a permanent are removed in pairs of a +1/+1 and -1/-1 counter as a state-based action (C.R. 704.5q), which happens any time a player would get priority (C.R. 117.5). The only moment it can happen that you proliferate (C.R. 701.27a) while a permanent has +1/+1 and -1/-1 counters on it is if, before you proliferate—
EDIT (Nov. 11): One rule was moved and spread out in Commander 2016.
EDIT (Apr. 5): Correction; added rule citations.
EDIT (May 12): Correctness edit.
EDIT (May 16): Edited post to conform to "put" rather than "place" convention introduced in Amonkhet.
EDIT (Sep. 28, 2017): One rule was renumbered with Ixalan.
EDIT (Nov. 29, 2017): Corrections to rule citations.
EDIT (Jan. 21, 2018): Add rule citation.
EDIT (May 3, 2019): Clarification.
EDIT (Jun. 8, 2019): Correctness edit.
EDIT (Dec. 25, 2019): Correctness edit. Some rules were renumbered with Core Set 2020.
EDIT (Oct. 24, 2022): Edited, including because some rules were renumbered in the meantime.